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Its Scope and Limits 



Ruskin House, Museum Street 


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T he following pages are addressed, not only or primarily 
to professional philosophers, but to that much larger public 
which is interested in philosophical questions without being 
willing or able to devote more than a limited amount of time 
to considering them. Descartes, Leibniz, Locke, Berkeley, and 
Hume wrote for a public of this sort, and I think it is unfortunate 
that during the last hundred and sixty years or so philosophy 
has come to be regarded as almost as technical as mathematics. 
Logic, it must be admitted, is technical in the same way as 
mathematics is, but logic, I maintain, is not part of philosophy. 
Philosophy proper deals with matters of interest to the general 
educated public, and loses much of its value if only a few pro- 
fessionals can understand what is said. 

In this book I have sought to deal, as comprehensively as I 
am able, with a very large question: how comes it that human 
beings, whose contacts with the world are brief and personal 
and limited, are nevertheless able to know as much as they do 
know? Is the belief in our knowledge partly illusory? And, if 
not, what must we know otherwise than through the senses? 
Since I have dealt in earlier books with some parts of this problem, 
I am compelled to repeat, in a larger context, discussions of 
certain matters which I have considered elsewhere, but I have 
reduced such repetition to the minimum compatible with my 

One of the difficulties of the subject with which I am concerned 
is that we must employ words which are common in ordinary 
speech, such as “belief”, “truth”, “knowledge”, and “percep- 
tion”. Since these words, in their every-day uses, are vague and 
unprecise, and since no precise words are ready to hand by 
which to replace them, it is inevitable that everything said in 
the earlier stages of our inquiry should be unsatisfactory from 
the point of view that we hope to arrive at in the end. Our 
increase of knowledge, assuming that we are successful, is like 
that of a traveller approaching a mountain through a haze: at 
first only certain large features are discernible, 'and even they 
have indistinct boundaries, but gradually more detail becomes 
visible and edges become sharper. So, in our discussions, it is 


human knowledge: its scope and limits 

impossible first to clear up one problem and then proceed to 
another, for the intervening haze envelops all alike. At every 
stage, though one part of our problem may be in the focus of 
attention, all parts are more or less relevant. The different key 
words that we must use are all interconnected, and so long as 
some remain vague, others must, more or less, share this defect. 
It follows that what is said at first is liable to require emendation 
later VThe Prophet announced that if two texts of the Koran 
appeared inconsistent, the later text was to be taken as authorita- 
tive, and I should wish the reader to apply a similar principle in 
'interpreting what is said in this book. 

The book has been read in typescript by my friend and pupil 
Mr. C. K. Hill, and I am indebted to him for many valuable 
criticisms, suggestions, and emendations. Large parts of the 
typescript have also been read by Mr. Hiram J. McLendon, 
who has made a number of useful suggestions. 

Part III, Chapter IV, on “Physics and Experience”, is a 
reprint, with few alterations, of a little book with the above 
title, published by the Cambridge University Press, to whom I 
owe thanks for permission to reprint it. 





I Individual and Social Knowledge 17 

II The Universe of Astronomy 23 

III The World of Physics 29 

IV Biological Evolution 43 

V The Physiology of Sensation and Volition 51 

VI The Science of Mind 57 


I The Uses of Language 71 

II Ostensive Definition 78 

III Proper Names 87 

IV Egocentric Particulars 100 

V Suspended Reactions: Knowledge and Belief 109 

VI Sentences 119 

VII External Reference of Ideas and Beliefs 123 

VIII Truth: Elementary Forms 127 

IX Logical Words and Falsehood 136 

X General Knowledge 146 

XI Fact, Belief, Truth, and Knowledge 159 


I Knowledge of Facts and Knowledge of Laws 180 

II Solipsism 191 

III Probable Inference in Common-sense Practice 198 

IV Physics and Experience 21 1 

V Time in Experience / 226 

VI Space in Psychology 233 

VII Mind and Matter 240 


HUMAN knowledge: its scope and limits 




tPAGE 251 


Minimum Vocabularies 






Structure and Minimum Vocabularies 



Time, Public and Private 



Space in Classical Physics 






The Principle of Individuation 



Causal Laws 



Space-time and Causality 






Kinds of Probability 



Mathematical Probability 



The Finite-Frequency Theory 



The Mises-Reichenbach Theory 



Keynes’s Theory of Probability 



Degrees of Credibility 



Probability and Induction 






Kinds of Knowledge 



The Role of Induction 

45 i 


The Postulate of Natural Kinds 



Knowledge Transcending Experience 



Causal Lines 

47 i 


Structure and Causal Laws 









Summary of Postulates 



The Limits of Empiricism 


Index 528 



T he central purpose of this book is to examine the relation 
between individual experience and the general body of 
scientific knowledge. It is taken for granted that scientific 
knowledge, in its broad outlines, is to be accepted. Scepticism, 
while logically impeccable, is psychologically impossible, and 
there is an element of frivolous insincerity in any philosophy 
which pretends to accept it. Moreover, if scepticism is to be 
theoretically defensible it must reject all inferences from what 
is experienced; a partial scepticism, such as the denial of physical 
events experienced by no one, or a solipsism which allows events 
in my future or in my unremembered past, has no logical 
justification, since it must admit principles of inference which 
lead to beliefs that it rejects. 

Ever since Kant, or perhaps it would be more just to say 
ever since Berkeley, there has been what I regard as a mistaken 
tendency among philosophers to allow the description of the 
world to be influenced unduly by considerations derived from 
the nature of human knowledge. To scientific common sense 
(which I accept) it is plain that only an infinitesimal part of the 
universe is known, that there were countless ages during which 
there was no knowledge, and that there probably will be countless 
ages without knowledge in the future. Cosmically and causally, 
knowledge is an unimportant feature of the universe; a science 
which omitted to mention its occurrence might, from an im- 
personal point of view, suffer only from a very trivial imperfection. 
In describing the world, subjectivity is a vice. Kant spoke of 
"himself as having effected a “Copernican revolution”, but he 
would have been more accurate if he had spoken of a “Ptolemaic 
counter-revolution”, since he put Man back at the centre from 
which Copernicus had dethroned him. 

But when we ask, not “what sort of world do we live in?” 
but “how do we come by our knowledge about the world?” 
subjectivity is in order. What each man knows is, in an important 
sense, dependent upon his own individual experience: he knows 
what he has seen and heard, what he has read and what he has 
been told, and also what, from these data, he has been able to 
infer. It is individual, not collective, experience that is here in 



question, for an inference is required to pass from my data 
to the acceptance of testimony. If I believe that there is such a 
place as Semipalatinsk, I believe it because of things that have 
happened to me\ and unless certain substantial principles of 
inference are accepted, I shall have to admit that all these things 
might have happened to me without there being any such place. 

The desire to escape from subjectivity in the description of 
the world (which I share) has led some modern philosophers 
astray — at least so it seems to me — in relation to theory of 
knowledge. Finding its problems distasteful, they have tried to 
deny that these problems exists That da ta are private. And indi- 
vidual is a thesis which has been familiar since the time of 
Protagoras. This thesis has been denied because it has been 
thought, as Protagoras thought, that, if admitted, it must lead 
to the conclusion that all knowledge is private and individual. 

( For my part, while I admit the thesis, I deny the conclusion; 
how and why, the following pages are intended to show. 

In virtue of certain events in my own life, I have a number 
of beliefs about events that I do not experience — the thoughts 
and feelings of other people, the physical objects that surround 
me, the historical and geological past of the earth, and the remote 
regions of the universe that are studied in astronomy. For my 
part, I accept these beliefs as valid, apart from errors of detail. 
By this acceptance I commit myself to the view that there are 
valid processes of inference from events to other events — more 
particularly, from events of which I am aware without inference 
to events of which I have no such awareness. To discover what 
these processes are is a matter of analysis of scientific and 
common-sense procedure, in so far as such procedure is generally 
accepted as scientifically valid. 

Inference from a group of events to other events can only be 
justified if the world has certain characteristics which are not 
logically necessary. So far as deductive logic can show, any 
collection of events might be the whole universe; if, then, I am 
ever to be able to infer events, I must accept principles of inference 
which lie outside deductive logic. All inference from events to 
events demands some kind of interconnection between different 
occurrences. Such interconnection is traditionally asserted in 
the principle of causality or natural law. It is implied, as we 
shall find, in whatever limited validity may be assigned to 



induction by simple enumeration. But the traditional ways of 
formulating the kind of interconnection that must be postulated 
are in many ways defective, some being too stringent and some 
not sufficiently sdt To discover the minimum principles required toj 
justify scientific inferences is one ofthe mam^purposes of this book j 
' iris ’'^ifonimoilpace*' to say that the substantial inferences 
of science, as opposed to those of logic and mathematics, are only 
probable — that is to say, when the premisses are true and the 
inference correct, the conclusion is only likely to be true. It is 
therefore necessary to examine what is meant by “probability”. 
It will be found that there are two different concepts that may 
be meant. On the one hand, there is mathematical probability: if 
a class has n members, and m of them have a certain characteristic, 
the mathematical probability that an unspecified member of this 
class will have the characteristic in question is m/n. On the other 
hand, there is a wider and vaguer concept, which I call “degree of 
credibility”, which is the amount of credence that it is rational to 
assign to a more or less uncertain proposition. Both kinds of proba- 
bility are involved in stating the principles of scientific inference. 

The course of our inquiry, in broad outline, will be as follows. 

Part I, on the world of science, describes some of the main 
features of the universe which scientific investigation has made 
probable. This Part may be taken as setting the goal which 
inference must be able to reach, if our data and our principles 
of inference are to justify scientific practice. 

Part II, on language, is still concerned with preliminaries. 
These are mainly of two sorts. On the one hand, it is important 
to make clear the meanings of certain fundamental terms, such 
as “fact” and “truth”. On the other hand, it is necessary to 
examine the relation of sensible experience to empirical concepts 
such as “red”, “hard”, “metre”, or “second”. In addition, we 
shall examine the relation of words having an essential reference 
to the speaker, such as “here” and “now”, to impersonal words, 
such as those assigning latitude, longitude, and date. This raises 
problems, of considerable importance and some difficulty, which 
are concerned with the relation of individual experience to the 
socially recognized body of general knowledge. 

In Part III, on Science and Perception, we begin our main 
inquiry. We are concerned, here, to disentangle data from 
inferences in what ordinarily passes for empirical knowledge. 


HUMAN knowledge: its scope and limits 

We are not yet concerned to justify inferences, or to investigate 
the principles according to which they are made, but we are 
concerned to show that inference s (as opposed to logica l Xr ^ 
constructio ns) are necessary to science. We are concerned also 
to distinguish between two kinds of space and time, one sub- 
jective and appertaining to data, the other objective and inferred. 
Incidentally we shall contend that solipsism, except in an extreme 
form in which it has never been entertained, is an illogical half- 
way house between the fragmentary world of data and the 
complete world of science. 

Part IV, on scientific concepts, is concerned to analyse the 
fundamental concepts of the inferred scientific world, more 
especially physical space, historical _time, and causal law s. The 
terms employed in mathematical physics are required to fulfil 
two kinds of conditions: on the one hand, they must satisfy 
certain formulae; on the other hand, they must be so interpreted 
as to yield results that can be confirmed or confuted by observa- 
tion. Through the latter condition they are linked to data, though 
somewhat loosely; through the former they become determinate 
as regards certain structural properties. But considerable latitude 
of interpretation remains. It is prudent to use this latitude in 
such a way as to minimize the part played by inference as 
opposed to construction; on this ground, for example, point- 
instants in space-time are constructed as groups of events or of 
qualities. Throughout this Part the two concepts of space-time 
structure and causal chains assume a gradually increasing 
importance.ftAs Part III was concerned to discover w hat can 
be counted as data, so Part IV is concerned to set forth, in a 
general wav, what, if science is to be j ustified, mimi-he afrle 
to infer from our 5a tal 

Since it is Admitted that scientific inferences, as a rule, only 
confer probability on their conclusions, Part V proceeds to the 
examination of Probability. This term is capable of various 
interpretations, and has been differently defined by different 
authors. These interpretations and definitions are examined, and 
so are the attempts to connect induction with probability. In 
this matter the conclusion reached is, in the main, that advocated 
by Keynes: that inductions do not make their conclusions 
probable unless certain conditions are fulfilled, and that experience 
alone can never prove that these conditions are fulfilled. 



Part VI, on the postulates of scientific i nferen ce, endeavours 
to discover what are the” minimum as sumptions, ante rior^to 
experience, that ar e required to justify usTn inferring laws from 
a^collection oFdat a; and further, to inqinre~~ih~ whaF sense, if 
any, we can be said to know that These lis's^^ 

The main logical function tTiaf' t^^assumptiohs Tiave to fulfil 
is that of conferring a high probability on the conclusions of 
inductions that satisfy certain conditions. For this purpose, 
since only probability is in question, we do not need to assume 
that such-and-such a connection of events occurs always, but 
only that it occurs frequently. For example, one of the assumptions 
that appear necessary is that of separable causal chains, such as 
are exhibited by light-rays or sound-waves. This assumption 
can be stated as follows : when an event having a complex space- 
time structure occurs, it frequently happens that it is one of a 
train of events having the same or a very similar structure. (A 
more exact statement will be found in Chapter VI of this Part.) 
This is part of a wider assumption of regularity, or natural law, 
which, however, requires to be stated in more specific forms than 
is usual, for in its usual form it turns out to be a tautology. 

That scientific inference requires, for its validity, principles 
which experience cannot render even probable, is, I believe, 
an inescapable conclusion from the logic of probability. For 
empiricism, it is an awkward conclusion. But I think it can be 
rendered somewhat more palatable by the analysis of the concept 
of “knowledge” undertaken in Part II. “Knowledge”, in my 
opinion, is a much less precise concept than is generally thought, 
and has its roots more deeply embedded in unverbalized animal 
behaviour than most philosophers have been willing to admit. 
The logically basic assumptions to which our analysis leads us are 
psychologically the end of a long series of refinements which 
start from habits of expectation in animals, such as that what 
has a certain kind of smell will be good to eat. To ask, therefore, 
whether we “know” the postulates of scientific inference, is not 
so definite a question as it seems. The answer must be: in one 
sense, yes, in another sense, no; but in the sense in which “no” 
is the right answer we know nothing whatever, and “knowledge” 
in this sense is a delusive vision. The perplexities of philosophers 
are due, in a large measure, to their unwillingness to awaken 
from this blissful dream. 




Chapter I 


S cientific knowledge aims at being wholly impersonal, and 
tries to state what has been discovered by the collective 
intellect of mankind. In this chapter I shall consider how 
far it succeeds in this aim, and what elements of individual 
knowledge have to be sacrificed in order to achieve the measure 
of success that is possible. 

The community knows both more and less than the individual : 
it knows, in its collective capacity, all the contents of the 
Encyclopaedia and all the contributions to the Proceedings of 
learned bodies, but it does not know the warm and intimate 
things that make up the colour and texture of an individual life. 
When a man says “I can never convey the horror I felt on seeing 
Buchenwald” or “no words can express my joy at seeing the sea 
again after years in a prison camp”, he is saying something which 
is strictly and precisely true : he possesses, through his experience, 
knowledge not possessed by those whose experience has been 
different, and not completely capable of verbal expression. If he 
is a superb literary artist he may create in sensitive readers a state 
of mind not wholly unlike his own, but if he tries scientific methods 
the stream of his experience will be lost and dissipated in a dusty 

Language, our sole means of communicating scientific know- 
ledge, is essentially social in its origin and in its main functions. 
It is true that, if a mathematician were wrecked or a desert island 
with a note-book and a pencil, he would, in all likelihood, seek to 
make his solitude endurable by calculations using the language of 
mathematics; it is true also that a man may keep a diary which 
he intends to conceal from all eyes but his own. On a more every- 
day plane, most of us use words in solitary thinking. Nevertheless 
the chief purpose of language is communication, and to serve this 
purpose it must be public, not a private dialect invented by the 
speaker. It follows that what is most personal in each individual's 
experience tends to evaporate during the process of ' translation 
into language. What is more, the very publicity of language is in 
large part a delusion. A given form of words will usually be 



human knowledge: its scope and limits 

interpreted by competent hearers in such a way as to be true for 
all of them or false for all of them, but in spite of this it will not 
have the same meaning for all of them. Differences which do not 
affect the truth or falsehood of a statement are usually of little 
practical importance, and are therefore ignored, with the result 
that we all believe our private world to be much more like the 
public world than it really is. 

This is easily proved by considering the process of learning to 
understand language. There are two ways of getting to know 
what a word means : one is by a definition in terms of other words, 
which is called verbal definition ; the other is by frequently hearing 
the word when the object which it denotes is present, which is 
called ostensive definition. It is obvious that ostensive definition is 
alone possible in the beginning, since verbal definition pre- 
supposes a knowledge of the words used in the definiens. You can 
learn by a verbal definition that a pentagon is a plane figure with 
five sides, but a child does not learn in this way the meaning of 
every-day words such as “rain”, “sun”, “dinner”, or “bed”. 
These are taught by using the appropriate word emphatically 
while the child is noticing the object concerned. Consequently the 
meaning that the child comes to attach to the word is a product of 
his personal experience, and varies according to his circumstances 
and his sensorium. A child who frequently experiences a mild 
drizzle will attach a different idea to the word “rain” from that 
formed by a child who has only experienced tropical torrents. A 
short-sighted and a long-sighted child will connect different 
images with the word “bed”. 

It is true that education tries to depersonalize language, and 
with a certain measure of success. “Rain” is no longer the familiar 
phenomenon, but “drops of water falling from clouds towards the 
earth”, and “water” is no longer what makes you wet, but H 2 0. 
As for hydrogen and oxygen, they have verbal definitions which 
have to be learnt by heart ; whether you understand them does not 
matter. And so, as your instruction proceeds, the world of words 
becomes more and more separated from the world of the senses ; 
you acquire the art of using words correctly, as you might acquire 
the art of playing the fiddle; in the end you become such a 
virtuoso in the manipulation of phrases that you need hardly ever 
remember that words have meanings. You have then become 
completely a public character, and even your inmost thoughts are 



suitable for the encyclopaedia. But you can no longer hope to be 
a poet, and if you try to be a lover you will find your depersonalized 
language not very successful in generating the desired emotions. 
You have sacrificed expression to communication, and what you 
can communicate turns out to be abstract and dry. 

It is an important fact that the nearer we come to the complete 
abstractness of logic, the less is the unavoidable difference between 
different people in the meaning attached to a word. I see no 
reason why there should be any difference at all between two 
suitably educated persons in the idea conveyed to them by the 
word “3481”. The words “or” and “not” are capable of having 
exactly the same meaning for two different logicians. Pure 
mathematics, throughout, works with concepts which are capable 
of being completely public and impersonal. The reason is that 
they derive nothing from the senses, and that the senses are the 
source of privacy. The body is a sensitive recording instrument, 
constantly transmitting messages from the outside world; the 
messages reaching one body are never quite the same as those 
reaching another, though practical and social exigencies have 
taught us ways of disregarding the differences between the 
percepts of neighbouring persons. In constructing physics we 
have emphasized the spatio-temporal aspect of our perceptions, 
which is the aspect that is most abstract and most nearly akin to 
logic and mathematics. This we have done in the pursuit of 
publicity, in order to communicate what is communicable and to 
cover up the rest in a dark mantle of oblivion. 

Space and time, however, as human beings know them, are not 
in reality so impersonal as science pretends. Theologians conceive 
God as viewing both space and time from without, impartially, 
and with a uniform awareness of the whole; science tries to 
imitate this impartiality with some apparent success, but the 
success is in part illusory. Human beings differ from the 
theologians* God in the fact that their space and time have a 
here and now. What is here and now is vivid, what is remote has 
a gradually increasing dimness. All our knowledge of events 
radiates from a space-time centre, which is the little region that 
we are occupying at the moment. “Here” is a vague term: in 
astronomical cosmology the Milky Way may count as “here”, in 
the study of the Milky Way “here” is the solar system, in the 
study of the solar system “here” is the earth, in geography it is 


human knowledge: its scope and limits 

the town or district in which we live, in physiological studies of 
sensation it is the brain as opposed to the rest of the body. Larger 
“heres” always contain smaller ones as parts; all “heres” contain 
the brain of the speaker, or part of it. Similar considerations apply 
to “now”. 

Science professes to eliminate “here” and “now”. When some 
event occurs on the earth's surface, we give its position in the 
space-time manifold by assigning latitude, longitude, and date. 
We have developed a technique which insures that all accurate 
observers with accurate instruments will arrive at the same 
estimate of latitude, longitude, and date. Consequently there is no 
longer anything personal in these estimates, in so far as we are 
content with numerical statements of which the meaning is not 
too closely investigated. Having arbitrarily decided that the 
longitude of Greenwich and the latitude of the equator are to 
be zero, other latitudes and longitudes follow. But what is 
“Greenwich”? This is hardly the sort of term that ought to 
occur in an impartial survey of the universe, and its definition is 
not mathematical. The best way to define “Greenwich” is to take 
a man to it and say: “Here is Greenwich.” If some one else has 
already determined the latitude and longitude of the place where 
you are, “Greenwich” can be defined by its latitude and longitude 
relative to that place; it is, for example, so many degrees east 
and so many degrees north of New York. But this does not get 
rid of “here”, which is now New York instead of Greenwich. 

Moreover it is absurd to define either Greenwich or New York 
by its latitude and longitude. Greenwich is an actual place, 
inhabited by actual people, and containing buildings which ante- 
date its longitudinal pre-eminence. You can, of course, describe 
Greenwich, but there always might be another town with the 
same characteristics. If you want to be sure that your description 
applies to no other place, the only way is to mention its relation 
to some other place, for instance, by saying that it is so many 
miles down the Thames from London Bridge. But then you will 
have to define “London Bridge”. Sooner or later you are faced 
with the necessity of defining some place as “here”, and this is 
an egocentric definition, since the place in question is not “here” 
for everybody. There may be a way of escape from this con- 
clusion; at a later stage, we will resume the question. But there 
is no obvious or easy way of escape, and until one is found all 



determinations of latitude and longitude are infected with the 
subjectivity of “here”. This means that, although different people 
assign the same latitude and longitude to a place, they do not, in 
ultimate analysis, attach the same meaning to the figures at which 
they arrive. 

The common world in which we believe ourselves to live is a 
construction, partly scientific, partly pre-scientific. We perceive 
tables as circular or rectangular, in spite of the fact that a painter, 
to reproduce their appearance, has to paint ellipses or non- 
rectangular quadrilaterals. We see a person as of about the same 
size whether he is two feet from us or twelve. Until our attention 
is drawn to the facts, we are quite unconscious of the corrections 
that experience has led us to make in interpreting sensible appear- 
ances. There is a long journey from the child who draws two eyes 
in a profile to the physicist who talks of electrons and protons, 
but throughout this journey there is one constant purpose: to 
eliminate the subjectivity of sensation, and substitute a kind of 
knowledge which can be the same for all percipients. Gradually 
the difference between what is sensed and what is believed to be 
objective grows greater; the child’s profile with two eyes is still 
very like what is seen, but the electrons and protons have only a 
remote resemblance of logical structure. The electrons and 
protons, however, have the merit that they may be what actually 
exists where there are no sense-organs, whereas our immediate 
visual data, owing to their subjectivity, are almost certainly not 
what takes place in the physical objects that we are said to see. 

The electrons and protons — assuming it scientifically correct to 
believe in them — do not depend for their existence upon being 
perceived; on the contrary, there is every reason to believe that 
they existed for countless ages before there were any percipients 
in the universe. But although perception is not needed for their 
existence, it is needed to give us a reason for believing in their 
existence. Hundreds of thousands of years ago, a vast and remote 
region emitted incredible numbers of photons, which wandered 
through the universe in all directions. At last a very few of them 
hit a photographic plate, in which they caused chemical changes 
which made parts of the plate look black instead of whjte when 
examined by an astronomer. This tiny effect upon a minute but 
highly educated organism is our only reason for believing in the 
existence of a nebula comparable in size with the Milky Way. 


human knowledge: its scope and limits 

ijfer The order for knowled ge is the inverse of the causal order. In the 
order" for knowledge, what comes first is the brief subjective 
experience of the astronomer looking at a pattern of black and 
white, and what comes last is the nebula, vast, remote, and 
belonging to the distant past. 

In considering the reasons for believing in any empirical 
statement, we cannot escape from perception with all its personal 
limitations. How far the information which we obtain from this 
tainted source can be purified in the filter of scientific method, 
and emerge resplendently godlike in its impartiality, is a difficult 
question, with which we shall be much concerned. But there is 
one thing that is obvious from the start: only in so far as the 
initial perceptual datum is trustworthy can there be any reason 
for accepting the vast cosmic edifice of inference which is based 
upon it. 

I am not suggesting that the initial perceptual datum must be 
accepted as indubitable ; that is by no means the case. There are 
well-known methods of strengthening or weakening the force of 
individual testimony; certain methods are used in the law courts, 
somewhat different ones are used in science. But all depend upon 
the principle that some weight is to be attached to every piece of 
testimony, for it is only in virtue of this principle that a number 
of concordant testimonies are held to give a high probability. 
Individual percepts are the basis of all our knowledge, and no 
method exists by which we can begin with data which are public 
to many observers. 


Chapter II 


Jk stronomy is the oldest of the sciences, and the contempla- 

L\ tion of the heavens, with their periodic regularities, gave 
A. Vmen their first conceptions of natural law. But in spite of 
its age, astronomy is as vigorous as at any former time, and as 
important in helping us to form a just estimate of man’s position 
in the universe. 

When the Greeks began inventing astronomical hypotheses, the 
apparent motions of the sun and moon and planets among the 
fixed stars had already been observed for thousands of years by 
the Babylonians and Egyptians, who had also learned to predict 
lunar eclipses with certainty and solar eclipses with a considerable 
risk of error. The Greeks, like other ancient nations, believed the 
heavenly bodies to be gods, or at any rate each closely controlled 
by its own god or goddess. Some, it is true, questioned this opinion : 
Anaxagoras, in the time of Pericles, maintained that the sun was 
a red-hot stone and that the moon was made of earth. But for this 
opinion he was prosecuted and compelled to fly from Athens. It 
is very questionable whether either Plato or Aristotle was equally 
rationalistic. But it was not the most rationalistic among the 
Greeks who were the best astronomers; it was the Pythagoreans, 
to whom superstition suggested what happened to be good 

The Pythagoreans, towards the end of the fifth century b.c., 
discovered that the earth is spherical; about a hundred years 
later, Eratosthenes estimated the earth’s diameter correctly within 
about fifty miles. Heraclides of Pontus, during the fourth century, 
maintained that the earth rotates once a day and that Venus and 
Mercury describe orbits about the sun. Aristarchus of Samos, in 
the third century, advocated the complete Copernican system, 
and worked out a theoretically correct method of estimating the 
distances of the sun and moon. As regards the sun this result, it 
is true, was wildly wrong, owing to inaccuracy in his data; but a 
hundred years later Posidonius made an estimate which was 
about half of the correct figure. This extraordinarily vigorous 
advance, however, did not continue, and much of it was for- 


human knowledge: its scope and limits 

gotten in the general decay of intellectual energy during later 

The cosmos, as it appears, for instance, in Plotinus, was a cosy 
and human little abode in comparison with what it has since 
become. The supreme deity regulated the whole, but each star 
was a subordinate deity, similar to a human being but in every 
way nobler and wiser. Plotinus finds fault with the Gnostics for 
believing that, in the created universe, there is nothing more 
worthy of admiration than the human soul. The beauty of the 
heavens, to him, is not only visual, but also moral and intellectual. 
The sun and moon and planets are exalted spirits, actuated by 
such motives as appeal to the philosopher in his best moments. 
He rejects with indignation the morose view of the Gnostics (and 
later of the Manicheans) that the visible world was created by a 
wicked Demiurge and must be despised by every aspirant to true 
virtue. On the contrary, the bright beings that adorn the sky are 
wise and good, and such as to console the philosopher amid the 
welter of folly and disaster that was overtaking the Roman Empire. 

The medieval Christian cosmos, though less austere than that 
of the Manicheans, was shorn of some elements of poetic fancy 
that paganism had preserved to the end. The change, however, 
was not very great, for angels and archangels more or less took 
the place of the polytheists’ celestial divinities. Both the scientific 
and the poetic elements of the medieval cosmos are set forth in 
Dante’s Paradiso ; the scientific elements are derived from 
Aristotle and Ptolemy. The earth is spherical, and at the centre 
of the universe ; Satan is at the centre of the earth, and hell is an 
inverted cone of which he forms the apex. At the antipodes of 
Jerusalem is the Mount of Purgatory, at whose summit is the 
earthly paradise, which is just in contact with the sphere of the 

The heavens consist of ten concentric spheres, that of the moon 
being the lowest. Everything below the moon is subject to cor- 
ruption and decay; everything from the moon upwards is in- 
destructible. Above the moon, the spheres in their order are those 
of Mercury, Venus, the Sun, Mars, Jupiter, Saturn and the fixed 
stars, beyond which is the Primum Mobile. Last of all, above the 
Primum Mobile, is the Empyrean, which has no motion, and in 
which there are no times or places. God, the Aristotelian Unmoved 
Mover, causes the rotation of the Primum Mobile, which, in turn, 



communicates its motion to the sphere of the fixed stars, and so 
on downwards to the sphere of the moon. Nothing is said in 
Dante as to the sizes of the various spheres, but he is able to 
traverse them all in the space of twenty-four hours. Clearly the 
universe as he conceived it was somewhat minute by modern 
standards; it was also very recent, having been created a few 
thousand years ago. The spheres, which all had the earth at the 
centre, afforded the eternal abodes of the elect. The elect consisted 
of those baptized persons who had reached the required standard 
both in faith and works, together with the patriarchs and prophets 
who had foreseen the coming of Christ, and a very few pagans 
who, while on earth, had been miraculously enlightened. 

It was against this picture of the universe that the pioneers of 
modern astronomy had to contend. It is interesting to contrast the 
commotion about Copernicus with the almost complete oblivion 
that befell Aristarchus. Cleanthes the Stoic had urged that 
Aristarchus should be prosecuted for impiety, but the Govern- 
ment was apathetic; perhaps if he had been persecuted, like 
Galileo, his theories might have won wider publicity. There were, 
however, other more important reasons for the difference between 
the posthumous fame of Aristarchus and that of Copernicus. In 
Greek times astronomy was an amusement of the idle rich — a very 
dignified amusement, it is true, but not an integrated part of the 
life of the community. By the sixteenth century, science had 
invented gunpowder and the mariner’s compass, the discovery of 
America had shown the limitations of ancient geognosis, Catholic 
orthodoxy had begun to seem an obstacle to material progress, and 
the fury of obscurantist theologians made the men of science 
appear as heroic champions of a new wisdom. The seventeenth 
century, with the telescope, the science of dynamics, and the law 
of gravitation, completed the triumph of the scientific outlook, 
not only as the key to pure knowledge, but as a powerful means 
of economic progress. From this time onwards, science was 
recognized as a matter of social and not merely individual interest. 

The theory of the sun and planets as a finished system was 
practically completed by Newton. As against Aristotle and the 
medieval philosophers it appeared that the sun, not the earth, is 
the centre of the solar system; that the heavenly Bodies, left to 
themselves, would move in straight lines, not in circles; that in 
fact they move neither in straight lines nor in circles, but in 

2 5 

human knowledge: its scope and limits 

ellipses ; and that no action from outside is necessary to preserve 
their motion. But as regards the origin of the system Newton had 
nothing scientific to say; he supposed that at the Creation the 
planets had been hurled by the hand of God in a tangential 
direction, and had then been left by Him to the operation of the 
law of gravitation. Before Newton, Descartes had attempted a 
theory of the origin of the solar system, but his theory proved 
untenable. Kant and Laplace invented the nebular hypothesis, 
according to which the sun was formed by the condensation of a 
primitive nebula, and threw off the planets successively as a 
result of increasingly rapid rotation. This theory also proved 
defective, and modern astronomers incline to the view that the 
planets were caused by the passage of another star through the 
near neighbourhood of the sun. The subject remains obscure, but 
no one doubts that, by some mechanism, the planets came out of 
the sun. 

The most remarkable astronomical progress in recent times has 
been in relation to the stars and the nebulae. The nearest of the 
fixed stars, Alpha Centauri, is at a distance of about 25 X io 12 
miles, or 4-2 light-years. (Light travels 186,000 miles a second; 
a light-year is the distance it travels in a year.) The first deter- 
mination of the distance of a star was in 1835; since then, by 
various ingenious methods, greater and greater distances have been 
computed. It is believed that the most distant object that can be 
detected with the most powerful telescope now in existence is 
about 500 million light-years away. 

Something is now known of the general structure of the 
universe. The sun is a star in the galaxy, which is an assembly of 
about 300,000 million stars, about 150,000 light-years across and 
between 25,000 and 40,000 light-years thick. The total mass of 
the galaxy is about 160,000 million times the mass of the sun; 
the mass of the sun is about 2 X io 27 tons. The whole of this 
system is slowly rotating about its centre of gravity ; the sun takes 
about 225 million years to complete its orbit round the milky way. 

In the space beyond the milky way, other systems of stars, of 
approximately the same size as the milky way, are scattered at 
fairly regular intervals throughout the space that our telescopes 
can explore. These systems are called extra-galactic nebulae ; it is 
thought that about 30 millions of them are visible, but the census 
is not yet complete. The average distance between two nebulae is 



about 2 million light-years. (Most of these facts are taken from 
Hubble, The Realm of the Nebulae , 1936.) 

One of the oddest facts about the nebulae is that the lines in 
their spectra, with very few exceptions, are shifted towards the 
red, and that the amount of the shift is proportional to the distance 
of the nebula. The only plausible explanation is that the nebulae 
are moving away from us, and that the most distant ones are 
receding most quickly. At a distance of 135 million light-years, 
this velocity amounts to 14,300 miles per second (Hubble, Plate 
VIII, p. 1 18). At a certain distance, the velocity would become 
equal to the velocity of light, and the nebulae would therefore be 
invisible however powerful our telescopes might be. 

The general theory of relativity has an explanation to offer of 
this curious phenomenon. The theory maintains that the universe 
is of finite size — not that it has an edge, outside which there is 
something which is not part of the universe, but that it is a three- 
dimensional sphere, in which the straightest possible lines return 
in time to their starting-point, as on the surface of the earth. The 
theory goes on to predict that the universe must be either con- 
tracting or expanding; it then uses the observed facts about the 
nebulae to decide for expansion. According to Eddington, the 
universe doubles in size every 1,300 million years or so. (New 
Pathways in Science , p. 210.) If this is true, the universe was once 
quite small, but will in time become rather large. 

This brings us to the question of the ages of the earth and the 
stars and the nebulae. On grounds that are largely geological, the 
age of the earth is estimated at about 3,000 million years. The age 
of the sun and the other stars is still a matter of controversy. If, 
in the interior of a star, matter can be annihilated by transforming 
an electron and a proton into radiation, the stars may be several 
million million years old; if not, only a few thousand million. 
(H. Spencer Jones, Worlds Without End , p. 231.) On the whole, 
the latter view seems to be prevailing. 

There is even some reason to think that the universe had a 
beginning in time; Eddington used to maintain that it began in 
about 90,000 million B.c, This is certainly more than the 4,004 
in which our great-grandfathers believed, but it is still a finite 
period, and raises all the old puzzles as to what was going on 
before that date. 

The net result of this summary survey of the astronomical 


HUMAN knowledge: its scope and limits 

When the forces to which a body is subject are not constant, the 
principle does not allow us to take each separately for a finite 
time, but if the finite time is short the result of taking each 
separately will be approximately right, and the shorter the time 
the more nearly right it will be, approaching complete rightness 
as a limit. 

It must be understood that this law is purely empirical ; there 
is no mathematical reason for its truth. It is to be believed in so 
far as there is evidence for it, and no further. In quantum 
mechanics it is not assumed, and there are phenomena which 
seem to show that it is not true in atomic occurrences. But in the 
physics of large-scale occurrences it remains true, and in classical 
physics it played a very important role. 

From Newton to the end of the nineteenth century, the progress 
of physics involved no basically new principles. The first revo- 
lutionary novelty was Planck's introduction of the quantum 
constant h in the year 1900. But before considering quantum 
theory, which is chiefly important in connection with the structure 
and behaviour of atoms, a few words must be said about 
relativity, which involved a departure from Newtonian principles 
much slighter than that of quantum theory. 

^Newton believed that, in addition to matter, there is absolute 
space and absolute time. That is to say, there is a three- dimensional 
manifold of points and a one-dimensional manifold of instant s, 
and there is~^TRree-term relation involving matter* space, and 
timejliamely the relation of “occupying” a point at an instant. 
In this view Newton agreed with Democritus and the other 
atomists of antiquity, who believed in “atoms and the void". 
Other philosophers had maintained that empty space is nothing, 
and that there must be matter everywhere. This was Descartes' 
opinion, and also that of Leibniz, with whom Newton (using Dr. 
Clarke as his mouthpiece) had a controversy on the subject. 

Whatever physicists might hold as a matter of philosophy, 
Newton’s view was implicit in the technique of dynamics, and 
there were, as he pointed out, empirical reasons for preferring it. 
If water in a bucket is rotated, it climbs up the sides, but if the 
bucket is rotated while the water is kept still, the surface of the 
water remains flat. We can therefore distinguish between rotation 
of the water and rotation of the bucket, which we ought not to be 
able to do if rotation were merely relative. Since Newton's time 

3 * 


other arguments of the same sort have accumulated. Foucault’s 
pendulum, the flattening of the earth at the poles, and the fact 
that bodies weigh less in low latitudes than in high ones, would 
enable us to infer that the earth rotates even if the sky were always 
covered with clouds; in fact, on Newtonian principles we can say 
that the rotation of the earth, not the revolution of the heavens, 
causes the succession of night and day and the rising and setting 
of the stars. But if space is purely relative, the difference between 
the statements “the earth rotates” and “the heavens revolve” is 
purely verbal : both must be ways of describing the same pheno- 

^Einstein showed how to avoid Newton’s conclusions, and make 
spatio-temporal position purely relative. But his theory of 
relativity did much more than this. In the special theory of 
relativity he showed that between two events there is a relation, 
which may be called “interval”, which can be divided in many 
different ways into what we should regard as a spatial distance 
and what we should regard as a lapse of time. All these different 
ways are equally legitimate ; there is not one way which is more 
“right” than the others. The choice between them is a matter of 
pure convention, like the choice between the metric system and 
the system of feet and inches !j 

It follows from this that the fundamental manifold of physics 
cannot consist of persistent particles in motion, but must consist 
of a four-dimensional manifold of “events’*? There will be three 
co-ordinates to fix the position of the event in space, and one to 
fix its position in time, but a change of co-ordinates may alter the 
time-co-ordinate as well as the space co-ordinates, and not only, 
as before, by a constant amount, the same for all events — as, for 
example, when dating is altered from the Mohammedan era to 
the Christian. 

The general theory of relativity — published in 1915, ten years 
after the special theory — was primarily a geometrical theory of 
gravitation. This part of the theory may be considered firmly 
established. But it has also more speculative features. It contains, 
in its equations, what is called the “cosmical constant”, which 
determines the size of the universe at any time. This part of the 
theory, as I mentioned before, is held to show that the universe 
is growing either continually larger or continually smaller.^ The 
shift towards the red in the spectra of distant nebulae is held to 



human knowledge: its scope and limits 

show that they are moving away from us with a velocity pro- 
portional to their distance from us. This leads to the conclusion 
that the universe is expanding, not contracting. It must be under- 
stood that, according to this theory, the universe is finite but 
unbounded, like the surface of a sphere, but in three dimensions. 
All this involves non-Euclidean geometry, and is apt to seem 
mysterious to those whose imagination is obstinately Euclidean. 

Two kinds of departure from Euclidean space are involved in 
the general theory of relativity. On the one hand, there are what 
may be called the small-scale departures (where the solar system, 
e.g., is regarded as “small”), and on the other hand the large-scale 
departure of the universe as a whole. The small-scale departures 
occur in the neighbourhood of matter, and account for gravitation. 
They may be compared to hills and valleys on the surface of the 
earth. The large-scale departure may be compared with the fact 
that the earth is round and not flat. If you start from any point 
on the earth’s surface and travel as straight as you can, you will 
ultimately return to your starting-point. So, it is held, the 
straightest line possible in the universe will ultimately return into 
itself. The analogy with the surface of the earth fails in that the 
earth’s surface is two-dimensional and has regions outside it, 
whereas the spherical space of the universe is three-dimensional 
and has nothing outside it. The present circumference of the 
universe is between 6,000 and 60,000 million light-years, but the 
size of the universe is doubled about every 1,300 million years. 
All this, however, must still be regarded as open to doubt. 

According to Professor E. A. Milne, 1 there is a great deal more 
that is questionable in Einstein’s theory. Professor Milne holds 
that there is no need to regard space as non-Euclidean, and that 
the geometry we adopt can be decided entirely by motives of 
convenience. The difference between different geometries, accord- 
ing to him, is a difference in language, not in what is described. 
Where physicists disagree it is rash for an outsider to have an 
opinion, but I incline to think that Professor Milne is very likely 
to be in the right. 

^. Quantum theory , i n co ntrast to the theory of relativity, is 
mainly concerned with the smallest things about which knowledge 
is pdssiBTe, Tiamely atoms and their structure. During the nine- 

1 Relativity Gravitation qnd World Structure . By E. A. Milne. Oxford, 

* 935 - 



teenth century the atomic constitution of matter became well 
established, and it was found that the different elements could be 
placed in a series starting with hydrogen and ending with uranium. 
The place of an element in this series is called its “atomic 
number”. Hydrogen has the atomic number i, and uranium 92. 
There are two gaps in the series at present, so that the number of 
known elements is 90, not 92; but the gaps may be filled any day, 
as a number of previously existing gaps have been. In general, but 
not always, the atomic number increases with the atomic weight. 
Before Rutherford, there was no plausible theory as to the structure 
of atoms, or as to the physical properties which caused them to 
fall into a series. The series was determined by their chemical 
properties alone, and of these properties no physical explanation 

The Rutherford-Bohr atom, as it is called after its two inventors, 
had a beautiful simplicity, now, alas, lost. But although it has 
become only a pictorial approximation to the truth, it can still be 
used when extreme accuracy is not required, and without it the 
modern quantum theory could never have arisen. It is therefore 
still necessary to say something about it. 

Rutherford gave experimental reasons for regarding an atom as 
composed of a nucleus carrying positive electricity surrounded by 
very much lighter bodies, called “electrons”, which carried 
negative electricity, and revolved, like planets, in orbits about the 
nucleus. When the atom is not electrified, the number of planetary 
electrons is the atomic number of the element concerned ; at all times, 
the atomic number measures the net positive electricity carried 
by the nucleus. The hydrogen atom consists of a nucleus and one 
planetary electron; the nucleus of the hydrogen atom is called a 
“proton”. It was found that the nuclei of other elements could be 
regarded as composed of protons and electrons, the number of 
protons being greater than that of the electrons by the atomic 
number of the element. Thus helium, which is number 2, has a 
nucleus consisting of four protons and two electrons. The atomic 
weight is practically determined by the number of protons, since 
a proton has about 1,850 times the mass of an electron, so that the 
contribution of the electrons to the total mass is almost negligible. 

It has been found that, in addition to electrons and protons, 
there are two other constituents of atoms, which are called 
“positrons” and “neutrons”. A positron is just like an electron, 


HUMAN knowledge: its scope and limits 

except that it carries positive instead of negative electricity; it has 
the same mass as an electron, and probably the same size, in so 
far as either can be said to have a size. The neutron has no 
electricity, but has approximately the same mass as a proton. It 
seems not unlikely that a proton consists of a positron and a 
neutron. If so, there are three ultimate kinds of constituents in 
the perfected Rutherford-Bohr atom: the neutron, which has 
mass but no electricity, the positron, carrying positive electricity, 
and the electron, carrying an equal amount of negative electricity. 

But we must now return to theories which ante-date the 
discovery of neutrons and positrons. 

Bohr added to the Rutherford picture a theory as to the possible 
orbits of electrons, which, for the first time, explained the lines 
in the spectrum of an element. This mathematical explanation was 
almost, but not quite, perfect in the cases of hydrogen and 
positively electrified helium; in other cases the mathematics was 
too difficult, but no reason appeared to suppose that the theory 
would give wrong results if the mathematics could be worked out. 
His theory made use of Planck’s quantum constant A, concerning 
which a few words must be said. 

Planck, by studying radiation, proved that in a light or heat 
wave of frequency v the energy must be h.v or 2 h.v or 3 h.v or 
some other integral multiple of A . v , where A is “Planck’s con- 
stant”, of which the value in C.G.S. units is about 6-55 x 10“ 27 , 
and the dimensions are those of action, i.e. energy x time. Before 
Planck, it had been supposed that the energy of a wave could vary 
continuously, but he showed conclusively that this could not be 
the case. The frequency of waves is the number that pass a given 
point in a second. In the case of light, the frequency determines 
the colour; violet light has the highest frequency, red light the 
lowest. There are other waves of just the same kind as light- waves, 
but not having the frequencies that cause visual sensations of 
colour. Higher frequencies than those of violet light are, in order, 
ultra-violet, X-rays and y-rays; lower frequencies, infra-red and 
those used in wireless telegraphy. 

When an atom emits light, it does so because it has parted with 
an amount of energy equal to that in the light-wave. If it emits 
light of frequency v , it must, according to Planck’s theory, have 
parted with an amount, of energy measured by A . v or some 
integral multiple of h.v. Bohr supposed that this happened through 



a planetary electron jumping from a larger to a smaller orbit; 
consequently the change of orbit must be such as to involve a loss 
of energy h . v or some integral multiple of this amount. It followed 
that only certain orbits could be possible. In the hydrogen atom, 
there would be a smallest possible orbit, and the other possible 
ones would have 4, 9, 16, . . . times the radius of the minimum 
orbit. This theory, first propounded in 1913, was found to agree 
well with observation, and for a time won general acceptance. 
Gradually, however, it was found that there were facts which it 
could not explain, so that, though clearly a step on the way to the 
truth, it could no longer be accepted as it stood. The new and 
more radical quantum theory, which dates from 1925, is due in 
the main to two men, Heisenberg and Schrodinger. 

In the modern theory there is no longer any attempt to make 
an imaginative picture of the atom. An atom only gives evidence 
of its existence when it emits energy, and therefore experimental 
evidence can only be of changes of energy. The new theory takes 
over from Bohr the doctrine that the energy in an atom must have 
one of a discrete series of values involving h; each of these is 
called an “energy lever’. But as to what gives the atom its energy 
the theory is prudently silent. 

One of the oddest things about the theory is that it has abolished 
the distinction between waves and particles. Newton thought that 
light consisted of particles emitted by the source of the light; 
Huygens thought that it consisted of waves. The view of Huygens 
prevailed, and until recently was thought to be definitely estab- 
lished. But new experimental facts seemed to demand that light 
should consist of particles, which were called “photons”. Per 
contra, De Broglie suggested that matter consists of waves. In 
the end it was shown that everything in physics can be explained 
either on the particle hypothesis or on the wave hypothesis. 
There is therefore no physical difference between them, and 
either may be adopted in any problem as may suit our convenience. 
But whichever is adopted, it must be adhered to; we must not 
mix the two hypotheses in one calculation. 

In quantum theory, individual atomic occurrences are not 
determined by the equations ; these suffice only to show that the 
possibilities form a discrete series, and that there' are rules 
determining how often each possibility will be realized in a large 
number of c*ses. There are reasons for believing that this absence 


HUMAN knowledge: its scope and limits 

of complete determinism is not due to any incompleteness in the 
theory, but is a genuine characteristic of small-scale occurrences. 
The regularity which is found in macroscopic phenomena is a 
statistical regularity. Phenomena involving large numbers of 
atoms remain deterministic, but what an individual atom may do 
in given circumstances is uncertain, not only because our 
knowledge is limited, but because there are no physical laws 
giving a determinate result. 

There is another result of quantum theory, about which, in 
my opinion, too much fuss has been made, namely what is called 
Heisenberg’s uncertainty-principle. According to this there is a 
theoretical limit to the accuracy with which certain connected 
quantities can be simultaneously measured. In specifying the state 
of a physical system, there are certain pairs of connected quantities ; 
one such pair is position and momentum (or velocity, so long 
as the mass is constant), another is energy and time. It is of course 
a commonplace that no physical quantity can be measured with 
complete accuracy, but it had always been supposed that there 
was no theoretical limit to the increase of accuracy obtainable 
by improved technique. According to Heisenberg’s principle this 
is not the case. If we try to measure simultaneously two con- 
nected quantities of the above sort, any increase of accuracy in 
the measurement of one of them (beyond a certain point) involves 
a decrease in the accuracy of the measurement of the other. In 
fact, there will be errors in both measurements, and the product 
of these two errors can never be less than hjzTT. This means that, 
if one could be completely accurate, the error in the other would 
have to be infinite. Suppose, for instance, that you wish to deter- 
mine the position and velocity of a particle at a certain time: if 
you get the position very nearly right, there will be a large error 
in the velocity, and if you get the velocity very nearly right, there 
will be a large error as to the position. Similarly as regards energy 
and time: if you measure the energy very accurately, the time 
when the system has this energy will have a large margin of un- 
certainty, while if you fix the time very accurately the energy will 
become uncertain within wide limits. This is not a question of 
imperfection in our measuring apparatus, but is an essential 
principle of physics. 

There are physical considerations which make this principle 
less surprising. It will be observed that h is a very small quantity, 



since it is of the order of io" 27 . Therefore wherever h is relevant 
we are concerned with matters involving very great minuteness. 
When an astronomer observes the sun, the sun preserves a lordly 
indifference to his proceedings. But when a physicist tries to find 
out what is happening to an atom, the apparatus by means of 
which he makes his observations is likely to have an effect upon 
the atom. Detailed considerations show that the sort of apparatus 
best suited for determining the position of an atom is likely to 
affect its velocity, while the sort of apparatus best suited for 
determining its velocity is likely to alter its position. Similar 
arguments apply to other pairs of related quantities. I do not 
think, therefore, that the uncertainty principle has the kind of 
philosophical importance that is sometimes attributed to it. 

Quantum equations differ from those of classical physics in 
a very important respect, namely that they are not “linear”. This 
means that when you have discovered the effect of one cause 
alone, and then the effect of another cause alone, you cannot find 
the effect of both together by adding the two previous effects. This 
has very odd results. Suppose, for instance, that you have a screen 
with a small slit, and you bombard it with particles; some of these 
will get through the slit. Suppose now you close the first slit and 
make a second ; then some will get through the second slit. Now 
open both slits at once. You would think that the number getting 
through both slits would be the sum of the previous numbers, 
but this turns out not to be the case. The behaviour of the par- 
ticles at one slit seems to be affected by the existence of the other 
slit. The equations are such as to predict this result, but it remains 
surprising. In quantum mechanics there is less independence of 
causes than in classical physics, and this adds greatly to the 
difficulty of the calculations. 

^Both relativity and quantum theory Jiave had the effect of 
replacing" the olef conception of “mass” by that of “energy”. 
“Mass” used to be defined as “quantity of matter”; “matter” 
was, on the one hand “substance” in the metaphysical sense, and 
on the other hand the technical form of the common-sense 
notion of “thing”. “Energy” was, in its early stages, a state of 
“matter”. It consisted of two parts, kinetic and potential. The 
kinetic energy of a particle is half the product of the' mass and 
the square of the velocity. The potential energy is measured by 
the work that would have to be done to bring the particle to its 


HUMAN knowledge: its scope and limits 

present position from some standard position. (This leaves a 
constant undetermined, but that is of no consequence.) If you 
carry a stone from the ground to the top of a tower, it acquires 
potential energy in the process; if you drop it from the top, the 
potential energy is gradually transformed into kinetic energy 
during the fall. In any self-contained system, the total energy is 
constant. There are various forms of energy, of which heat is one; 
there is a tendency for more and more of the energy in the universe 
to take the form of heat. The conservation of energy first became 
a well-grounded scientific generalization when Joule measured 
the mechanical equivalent of heat. 

Relativity theory and experiment both showed that mass is not 
constant, as had been held, but is increased by rapid motion; 
if a particle could move as fast as light, its mass would become 
infinite. Since all motion is relative, the different estimates of mass 
formed by different observers, according to their motion relative 
to the particle in question, are all equally legitimate. So far as this 
theory is concerned, however, there is still one estimate of mass 
which may be considered fundamental, namely the estimate made 
by an observer who is at rest relatively to the body whose mass 
is to be measured. Since the increase of mass with velocity is only 
appreciable for velocities comparable with that of light, this case 
covers practically all observations except those of a and j8 particles 
ejected from radio-active bodies. 

Quantum theory has made a greater inroad upon the concept 
of “mass”. It now appears that whenever energy is lost by radia- 
tion there is a corresponding loss of mass. The sun is held to 
be losing mass at the rate of four million tons a second. To take 
another instance: a helium atom, unelectrified, consists (in the 
language of Bohr’s theory) of four protons and four electrons, 
while a hydrogen atom consists of one proton and one electron. 
It might have been supposed that, assuming this to be the case, 
the mass of a helium atom would be four times that of a hydrogen 
atom. This, however, is not the case: taking the mass of the 
helium atom as 4, that of the hydrogen atom is not 1, but 1 -008. 
The reason is that energy is lost (by radiation) when four hydrogen 
atoms combine to form one helium atom — at least so we must 
suppose, for the process is not one which has ever been 

It is thought that the combination of four hydrogen atoms to 



form one atom of helium occurs in the interior of stars, and could 
be made to occur in terrestrial laboratories if we could produce 
temperatures comparable to those in the interior of stars. Almost 
all the loss of energy involved in building up elements other than 
hydrogen occurs in the transition to helium; in later stages the 
loss of energy is small. If helium, or any element other than 
hydrogen, could be artificially manufactured out of hydrogen, 
there would be in the process an enormous liberation of energy 
in the form of light and heat. This suggests the possibility of 
atomic bombs more destructive than the present ones, which are 
made by means of uranium. There would be a further advantage: 
the supply of uranium in the planet is very limited, and it is 
feared that it may be used up before the human race is exter- 
minated, but if the practically unlimited supply of hydrogen in 
the sea could be utilized there would be considerable reason to 
hope that homo sapiens might put an end to himself, to the great 
advantage of the other less ferocious animals. 

But it is time to return to less cheerful topics. 

The language of Bohr’s theory is still adequate for many pur- 
poses, but not for stating the fundamental principles of quantum 
physics. To state these principles, we must avoid all pictures of 
what goes on in an atom, and must abandon attempts to say what 
energy is. We must say simply: there is something quantitative, 
to which we give the name “energy”; this something is very 
unevenly distributed in space; there are small regions in which 
there is a great deal of it, which are called “atoms”, and are those 
in which, according to older conceptions, there was matter; these 
regions are perpetually absorbing or emitting energy in forms 
that have a periodic “frequency”. Quantum equations give rules 
determining the possible forms of energy emitted by a given atom, 
and the proportion of cases (out of a large number) in which each 
of the possibilities will be realized. Everything here is abstract 
and mathematical except the sensations of colour, heat, etc., 
produced by the radiant energy in the observing physicist. 

Mathematical physics contains such an immense superstructure 
of theory that its basis in observation tends to be obscured. It 
is, however, an empirical study, and its empirical character appears 
most unequivocally where the physical constants are concerned. 
Eddington {New Pathways in Science , p. 230) gives the following 
list of the primitive constants of physics : 


HUMAN knowledge: its scope and limits 

very different from the temperatures to which we are accustomed ; 
at these temperatures, molecules of a very high degree of com- 
plexity can come into existence. 

What distinguishes living from dead matter? Primarily, its 
chemical constitution and cell structure. It is to be supposed that 
its other characteristics follow from these. The most notable of 
these others are assimilation and reproduction, which, in the 
lowest forms of life, are not very sharply distinguished from each 
other. The result of assimilation and reproduction is that, given 
a small amount of living matter in a suitable environment, the 
total amount will quickly increase. A pair of rabbits in Australia 
quickly become many tons of rabbit. A few measles bacilli in 
a child quickly become many millions. A few seeds dropped by 
birds on Krakatoa after volcanic devastation quickly became 
luxuriant vegetation. So far as animals are concerned, this property 
of living matter is not fully exhibited, since animals require food 
that is already organic; but plants can transform inorganic sub- 
stances into living matter. This is a purely chemical process, but 
it is one from which, presumably, most of the other peculiarities 
of living matter, considered as a whole, in some sense follow. 

It is an essential feature of living matter that it is not chemically 
static, but is undergoing continual chemical change; it is, one 
may say, a natural chemical laboratory. Our blood undergoes one 
kind of change as it circulates round the body, and an opposite 
change when it comes in contact with air in the lungs. Food, 
from the moment of contact with the saliva, undergoes a series 
of elaborate processes, which end by giving it the chemical 
structure appropriate to some part of the body. 

There is no reason, except the great complexity of the molecules 
that compose a living body, why such molecules should not be 
manufactured artificially; nor is there the slightest reason for 
supposing that, if they were manufactured, they would lack any- 
thing distinctive of living matter naturally generated. Aristotle 
thought that there was a vegetable soul in every plant or animal, 
and something similar has been widely believed by vitalists. But 
for this view there has come to be less and less plausibility as 
organic chemistry has progressed. The evidence, though not 
conclusive, tends to show that everything distinctive of living 
matter can be reduced to chemistry, and therefore ultimately to 
physics. The fundamental laws governing living matter are, in 



all likelihood, the very same that govern the behaviour of the 
hydrogen atom, namely the laws of quantum mechanics. 

One of the characteristics of living organisms that have seemed 
mysterious is the power of reproduction. Rabbits generate rabbits, 
robins generate robins, and worms generate worms. Development 
from an embryo does not occur in the simplest forms of life; 
unicellular organisms merely grow till they reach a certain size, 
and then split. Something of this survives in sexual reproduction: 
part of the female body becomes an ovum, part of the male body 
a sperm, but this part is so much less than half that it seems 
qualitatively, and not merely quantitatively, different from the 
process of splitting into two equal halves. It is not in the splitting, 
however, that the novelty consists, but in the combination of male 
and female elements to make a new organism, which, in the 
natural process of growth, becomes, in time, like its adult parents. 

As a consequence of the Mendelian theory, the process of 
heredity has come to be more or less understood. It appears that 
in the ovum and in the sperm there are a certain fairly small 
number of “genes”, which carry the hereditary characteristics. 
The laws of heredity, like those of quantum theory, are discrete 
and statistical; in general, when grandparents differ in some 
character, we cannot tell which grandparent a given child will 
resemble, but we can tell the proportion, out of a large number, 
that will resemble this one or that as regards the character in 

In general, the genes carry the parental character, but some- 
times there are sports, or “mutants”, which differ substantially 
from the parent. They occur naturally in a small proportion of 
cases, and they can be produced artificially by X-rays. It is these 
sports that give the best opportunity for evolution, i.e. for the 
development of new kinds of animals or plants by descent from 
old kinds. 

The general idea of evolution is very old; it is already to be 
found in Anaximander (sixth century B.C.), who held that men 
are descended from fishes. But Aristotle and the Church banished 
such theories until the eighteenth century. Already Descartes, 
Kant, and Laplace had advocated a gradual origin for the solar 
system, in place of sudden creation followed by a complete absence 
of change. As soon as geologists had succeeded in determining 
the relative ages of different strata, it became evident from fossils 


HUMAN knowledge: its scope and limits 

planets. Life, therefore, is almost certainly a very rare phenomenon. 
Even on the earth it is transitory : at first the earth was too hot, 
and in the end it will be too cold. Some highly conjectural dates 
are suggested in Spencer Jones’s Worlds Without End (p. 19). The 
age of the earth is probably less than 3,000 million years; the 
beginnings of life may be placed at about 1,700 million years ago. 
Mammals began about 60 million years ago ; anthropoid apes about 
8 million, man about 1 million. It is probable that all forms of 
life on earth have evolved from unicellular organisms. How these 
were first formed we do not know, but their origin is no more 
mysterious than that of helium atoms. There is no reason to 
suppose living matter subject to any laws other than those to 
which inanimate matter is subject, and considerable reason to 
think that everything in the behaviour of living matter is theoreti- 
cally explicable in terms of physics and chemistry. 


Chapter V 


F rom the standpoint of orthodox psychology, there are two 
boundaries between the mental and physical, namely sen- 
sation and volition. (“Sensation” may be defined as the 
first mental effect of a physical cause^^^voTition^ as the last mental 
[caus e of a physical effect .yf am not maintaining that these 
definitions will prove ultimately satisfactory, but only that they 
may be adopted as a guide in our preliminary survey. In the 
present chapter I shall not be concerned with either sensation 
or volition themselves, since they belong to psychology; I shall 
be concerned only with (t he phy siological antecedents a nd con- 
comitants of sensation, and (with the physiological concomitants 
and" consequents of volition. Before considering what science’has 
tcTsayT irwttttye worth while to look at the matter first from a 
common-sense point of view. 

Suppose something is said to you, and in consequence you take 
some action; for example, you may be a soldier obeying the word 
of comman^Physics studies the sound waves that travel through 
the air until tney reach the ear ; physiology studies the consequent 
event in the ear and nerves and brain, up to the moment when 
you hear the sound ; psychology studies the sensation of hearing 
and the consequent volition; physiolo gy then resumes the study 
of the process, and considers the outgoing chain of events from 
the brain to the muscles and the bodily movement expressing 
the volition; from that point^pnward, what happens is again part 
of the subject-matter of physi cs. The problem of the relation of 
mind and matter, which is part of the stock in trade of philosophy, 
comes to a head in the transition from events in the brain to the 
sensation, jpid from the volition to other events in the brain. It 
is thus a two-fo ld probl em: (how does matter affect mind in 
sensation, ^and^iow does mind affect matter in volitional do not 
propose to consider this problem at this stage ; I mention it now 
only to show the relevance of certain parts of physiology to 
questions which philosophy must discuss. 

The physiological processes which precede and accompany 

5 * 

human knowledge: its scope and limits 

sensation are admirably set forth in Adrian’s book The Basis of 
Sensation: The Action of the Sense Organs (London, 1928). As 
every one knows, there are two sorts of nerve fibres, those that 
carry messages into the brain, and those that carry messages out 
of it. The former alone are concerned in the physiology of sen- 
sation. Isolated nerves can be stimulated artificially by an electric 
current, and there is good reason to believe that the processes 
thus set up are essentially similar to those set up naturally in 
nerves that are still in place in a living body. When an isolated 
nerve is thus stimulated in an adequate manner, a disturbance 
is set up which travels along the nerve at a speed of about 220 
miles an hour (100 metres a second). Each nerve consists of a 
bundle of nerve fibres running from the surface of the body to 
the brain or the spinal chord. The nerve fibres which carry mes- 
sages to the brain are called “afferen t”, those which carry messages 
from the brain are called “ effere nt”. A nerve usually contains both 
afferent and efferent fibres, firoadly speaking, the afferent fibres 
start from sense-organs and the efferent fibres end in muscles. 

The response of a nerve fibre to a stimulus is of what is called 
the “all-or-nothing” type, like the response of a gun to pressure 
on the trigger. A slight pressure on the trigger produces no result, 
but a pressure which is sufficiently great produces a specific result 
which is the same however great the pressure may be (within 
limits). Similarly when a nerve fibre is stimulated very slightly, 
or for a very brief period (less than • 00001 of a second), there 
is no result, but when the stimulus is sufficient a current travels 
along the nerve fibre for a very brief period (a few thousandths 
of a second), after which the nerve fibre is “tired” and will not 
transmit another current until it is rested. At first, for two or 
three thousandths of a second, the nerve fibre is completely 
refractory; then it recovers gradually. During the period of 
recovery a given stimulus produces a smaller response, and one 
which travels more slowly. Recovery is complete after about a 
tenth of a second. The result is that a constant stimulus does not 
produce a constant state of excitement in the nerve fibres, but 
a series of responses with quiescent periods between. The mes- 
sages that reach the brain are, as Adrian puts it, like a stream 
of bullets from a machine gun, not like a continuous stream of 

It is supposed that in the brain, or the spinal column, there 



is a converse mechanism which reconverts the discrete impulses 
into a continuous process, but this, so far, is purely hypothetical. 

Owing to the discontinuous nature of the response to a stimulus, 
the response will be exactly the same to a constant stimulus as to 
one which is intermittent with a frequency adapted to the period 
of recovery in the nerve. It would seem to follow that there can 
be no means of knowing whether the stimulus is constant or 
intermittent. But this is not altogether true. Suppose, for instance, 
that you are looking at a bright spot of light : if you could keep 
your eyes absolutely fixed, your sensations would be the same 
if the light flickered with appropriate rapidity as they would be 
if the light were steady. But in fact it is impossible to keep the 
eyes quite still, and therefore fresh unfatigued nerves are per- 
petually being brought into play. 

A remarkable fact, which might seem to put a limit on the 
informative value of sensations, is, that the response of the nerve 
fibre ip the same to any stimulus of sufficient strength and dura- 
tion: there is just one message, and only one, that a given nerve 
fibre can transmit. But consider the analogy of a typewriter: 
if you press a given letter, only one result occurs, and yet the 
typewriter as a whole can transmit any information, however 

The mechanism of the efferent nerve fibres appears to be just 
the same as that of the afferent nerve fibres ; the messages that 
travel from the brain to the muscles have the same jerky character 
as those that travel from the sense-organs to the brain. 

But the most interesting question remains: what goes on in 
the brain between the arrival of a message by the afferent nerves 
and the departure of a message by the efferent nerves ? Suppose 
you read a telegram saying “all your property has been destroyed 
in an earthquake”, and you exclaim “heavens! I am ruined”. 
We feel, rightly or wrongly, that we know the psychological links, 
after a fashion, by introspection, but everybody is agreed that 
there must also be physiological links. The current brought into 
the vision centre by the optic nerve must pass thence to the speech 
centre, and then stimulate the muscles which produce your 
exclamation. How this happens is still obscure. But it seems clear 
that, from a physiological point of view, there is a unitary process 
from the physical stimulus to the muscular response. In man this 
process may be rendered exceedingly complex by the operation 


HUMAN knowledge: its scope and limits 

they are taught by different men. What physicists have to teach 
is fairly clear, but what have the psychologists to teach ? 

There are those among psychologists who take a view which 
really denies to psychology the status of a separate science. Accord- 
ing to this school, psychology consists in the study of human 
and animal behaviour, and the only thing that distinguishes it 
from philosophy is its interest in the organism as a whole. The 
observations upon which the psychologist must rely, according to 
this view, are such as a man might make on animals other than 
himself; there is no science, say the adherents of this school, 
which has data that a man can only obtain by observation of him- 
self. While I admit the importance of what has been learnt by 
studying behaviour, I cannot accept this view. There are — and I 
am prepared to maintain this dogmatically — many kinds of events 
that I can observe when they happen to me, but not when they 
happen to any one else. I can observe my own pains and pleasures, 
my perceptions, my desires, my dreams. Analogy leads me to 
believe that other people have similar experiences, but this is an 
inference, not an observation. The dentist does not feel my 
toothache, though he may have admirable inductive grounds for 
believing that I do. 

This suggests a possible definition of psychology, as the science 
of those occurrences which, by their very nature, can only be 
observed by one person. Such a definition, however, unless 
somewhat limited, will turn out to be too wide in one direction, 
while too narrow in another. When a number of people observe a 
public event, such as the bursting of a rocket or a broadcast by 
the Prime Minister, they do not all see or hear exactly the same 
thing: there are differences due to perspective, distance from the 
source of the sight or sound, defects in the sense-organs, and so 
on. Therefore if we were to speak with pedantic accuracy, we 
should have to say that everything that can be observed is private 
to one person. There is often, however, such a close similarity 
between the simultaneous percepts of different people that the 
minute differences can, for many purposes, be ignored; we then 
say that they are all perceiving the same occurrence, and we place 
this occurrence in a public world outside all the observers. Such 
occurrences are the data of physics, while those that have not this 
social and public character supply (so I suggest) the data of 



According to this view, a datum for physics is something 
abstracted from a system of correlated psychological data. When 
a crowd of people all observe a rocket bursting, they will ignore 
whatever there is reason to think peculiar and personal in their 
experience, and will not realize without an effort that there is any 
private element in what they see. But they can, if necessary, 
become aware of these elements. One part of the crowd sees the 
rocket on the right, one on the left, and so on. Thus when each 
person's perception is studied in its fullness, and not in the 
abstract form which is most convenient for conveying information 
about the outside world, the perception becomes a datum for 

But although every physical datum is derived from a system of 
psychological data, the converse is not the case. Sensations 
resulting from a stimulus within the body will naturally not be 
felt by other people ; if I have a stomach-ache I am in no degree 
surprised to find that others are not similarly afflicted. There are 
afferent nerves from the muscles, which cause sensations when 
the muscles are used; these sensations, naturally, are only felt by 
the person concerned. It is only when the stimulus is outside the 
body of the percipient, and not always even then, that the sensa- 
tion is one of a system which together constitutes one datum for 
physics. If a fly is crawling on your hand, the visual sensations 
that it causes are public, but the tickling is private. Psychology is 
the science which deals with private data, and with the private 
aspects of data which common sense regards as public. 

To this definition a fundamental objection is raised by a certain 
school of psychologists, who maintain that “introspection" is not 
a valid scientific method, and that nothing can be scientifically 
known except what is derived from public data. This view seems 
to me so absurd that if it were not widely held I should ignore 
it; but as it has become fashionable in various circles I shall state 
my reasons for rejecting it. 

To begin with, we need a more precise definition of “public" 
and “private" data. “Public" data, for the purpose of those who 
reject introspection, are not only data which in fact are shared 
by other observers, but also those which might be so shared 
given suitable circumstances. Robinson Crusoe, on this view, is 
not being unscientifically introspective when he describes the 
crops he raised, although there is no other observer to confirm his 



narrative, for its later parts are confirmed by Man Friday, and 
its earlier parts might have been. But when he relates how he 
became persuaded that his misfortunes were a punishment for 
his previous sinful life, he is either saying something meaningless 
or telling what words he would have uttered if he had had any one 
to speak to — for what a man says is public, but what he thinks is 
private. To maintain that what he says expresses his thought is, 
according to this school, to say something not scientifically 
verifiable and therefore something which science should not say. 
To attempt — as Freud did — to make a science of dreams is a 
mistake; we cannot know what a man dreams, but only what he 
says he dreams. What he says he dreams is part of physics, 
since the saying consists of movements of lips and tongue and 
throat; but it is a wanton assumption to suppose that what he 
says in professing to relate his dream expresses an actual 

We shall have to define a “public” datum as one which can be 
observed by many people, provided they are suitably placed. 
They need not all observe it at once, provided there is reason to 
think that there has been no change meanwhile: two people 
cannot look down a microscope at the same time, but the enemies 
of introspection do not mean to exclude data obtained by means 
of microscopes. Or consider the fact that, if you press one eyeball 
upwards, everything looks double. What is meant by saying that 
things “look” double? This can only be interpreted by distinguish- 
ing between the visual perception and the physical fact, or else by 
a subterfuge. You may say: “When I say that Mr. A. is seeing 
double, I say nothing about his perceptions; what I say means: 
Tf Mr. A. is asked, he will say he is seeing double*.” Such an 
interpretation makes it meaningless to inquire whether Mr. A. is 
speaking the truth, and impossible to discover what it is that he 
thinks he is asserting. 

Dreams are perhaps the most indubitable example of facts 
which can only be known by means of private data. When I 
remember a dream I can relate it, either truly or with embellish- 
ments; I can know which I am doing, but others seldom can. I 
knew a Chinese lady who, after a few lessons in psycho-analysis, 
began to have perfect text-book dreams; the analyst was de- 
lighted, but her friends were sceptical. Although no one except 
the lady could be sure of the truth, I maintain that the fact as to 



what she had dreamed was just as definitely such-and-such rather 
than so-and-so as in the case of a physical phenomenon. 

We shall have to say: A “public” datum is one which generates 
similar sensations in all percipients throughout a certain space- 
time region, which must be considerably larger than the region 
occupied by one human body throughout (say) half a second — 
or rather, it is one which would generate such sensations if 
suitably placed percipients were present (this is to allow for 
Robinson Crusoe’s crops). 

This distinction between public and private data is one which 
it is difficult to make precise. Roughly speaking, sight and hearing 
give public data, but not always. When a patient is suffering from 
jaundice everything looks yellow, but this yellowness is private. 
Many people are liable to a buzzing in the ears which is sub- 
jectively indistinguishable from the hum of telegraph wires in a 
wind. The privacy of such sensations is only known to the 
percipient through the negative testimony of other people. Touch 
gives public data in a sense, since different people can successively 
touch the same object. Smells can be so public as to become 
grounds of complaint to the sanitary authority. Tastes are public 
in a lesser degree, for, though two people cannot eat the same 
mouthful, they can eat contiguous portions of the same viand ; but 
the curate’s egg shows that this method is not quite reliable. It 
is, however, sufficiently reliable to establish a public distinction 
between good cooks and bad ones, though here introspection 
plays an essential part, for a good cook is one who causes 
pleasure to most consumers, and the pleasure of each is purely 

I have kept this discussion on a common-sense level, but at a 
later stage I shall resume it, and try to probe more deeply into 
the whole question of private data as a basis for science. For the 
present I am content to say that the distinction between public 
and private data is one of degree, that it depends upon testimony 
which bears witness to the results of introspection, that physiology 
would lead to the expectation that sensations caused by a stimulus 
inside a human body would be private, and, finally, that many of 
the facts of which each one of us is most certain are known to us 
by means private to ourselves. Do you like the smell of rotten 
eggs ? Are you glad the war is over ? Have you a toothache ? These 
questions are not difficult for you to answer, but no one else can 


HUMAN knowledge: its scope and limits 

answer them except by inferences from your behaviour, including 
your testimony. 

I conclude, therefore, that there is knowledge of private data, 
and that there is no reason why there should not be a science of 
them. This being granted, we can now inquire what psychology 
in fact has to say. 

There is, to begin with, a matter of which the importance is 
often overlooked, and that is the correlation of physical occurrences 
with sensation. Physicists and astronomers base their assertions 
as to what goes on in the outer world upon the evidence of the 
senses, especially the sense of sight. But not a single one of the 
occurrences that we are told take place in the physical world is a 
sensation; how, then, can sensations confirm or confute a physical 
theory? Let us take an illustration belonging to the infancy of 
science. It was early discovered that an eclipse of the sun is due 
to the interposition of the moon, and it was found that eclipses 
could be predicted. Now what was directly verified when an 
eclipse occurred was a certain sequence of expected sensations. 
But the development of physics and physiology has gradually 
caused a vast gulf between the sensations of an astronomer watching 
an eclipse and the astronomical fact which he infers. Photons start 
from the sun, and when the moon is not in the way some of them 
reach an eye, where they set up the kind of complicated process 
that we considered in the last chapter; at last, when the process 
reaches the astronomer’s brain, the astronomer has a sensation. 

The sensation can only be evidence of the astronomical fact if 
laws are known connecting the two, and the last stage in these 
laws must be one connecting stimulus and sensation, or connecting 
occurrences in the optic nerve or the brain with sensation. The 
sensation, it should be observed, is not at all like the astronomical 
fact, nor are the two necessarily connected. It would be possible 
to supply an artificial stimulus causing the astronomer to have 
an experience subjectively indistinguishable from what we call 
“seeing the sun”. And at best the resemblance between the 
sensation and the astronomical fact cannot be closer than that 
between a gramophone record and the music that it plays, or 
between a library catalogue and the books that it enumerates. It 
follows that, if physics is an empirical science, whose statements 
can be confirmed or confuted by observation, then physics must 
be supplemented by laws connecting stimulus and sensation. 

6 z 


Now such laws belong to psychology. Therefore what is em- 
pirically verifiable is not pure physics in isolation, but physics 
plus a department of psychology. Psychology, accordingly, is an 
essential ingredient in every part of empirical science. 

So far, however, we have not inquired whether there are any 
laws that connect one mental event with another. The laws of 
correlation so far considered have been such as connect a physical 
stimulus with a mental response; what we have now to consider 
is whether there are any causal laws which are entirely within one 
mind. If there are, psychology is to that extent an autonomous 
science. The association of ideas, as it appears for example in 
Hartley and Bentham, was a law of this kind, but the conditioned 
reflex and the law of habit, which have taken its place, are 
primarily physiological and only derivatively psychological, since 
association is thought to be caused by the creation of paths in the 
brain connecting one centre with another. We may still state the 
association of ideas in purely psychological terms, but when so 
stated it is not a law as to what always happens, but only as to 
what is apt to happen. It has not therefore the character that 
science hopes to find in a causal law, or at least used to hope for 
before the rise of quantum theory. 

The same thing may be said of psycho-analysis, which aims 
at discovering purely mental causal laws. I do not know of any 
psycho-analytic law which professes to say what will always 
happen in such and such circumstances. When a man, for 
example, suffers from claustrophobia, psycho-analysis will 
discover this or that past experience which is held to explain 
his trouble; but many people will have had the same experience 
without the same result. The experience in question, accordingly, 
though it may well be part of the cause of the phobia, cannot 
be its whole cause. We cannot, this being the case, find in psycho- 
analysis any examples of purely psychical causal laws. 

In the last chapter we suggested, as a probable hypothesis, 
the view that all bodily behaviour is theoretically explicable in 
physical terms, without taking any account of the mental 
concomitants of physiological occurrences. This hypothesis, it 
should be observed, in no way decides our present question. 
If A and B are two events in the brain, and if A causes B, then 
if a is a mental concomitant of A, and b of B, it will follow that 
a causes A, which is a purely mental causal law. In fact, causal 


human knowledge: its scope and limits 

laws are not of the simple form “A causes B”, but in their true 
form the principle remains the same. 

Although, at present, it is difficult to give important examples 
of really precise mental causal laws, it seems pretty certain, on 
a common-sense basis, that there are such laws. If you tell a 
man that he is both a knave and a fool, he will be angry; if you 
inform your employer that he is universally regarded as a 
swindler and a bloodsucker, he will invite you to seek employment 
elsewhere. Advertising and political propaganda supply a mass 
of materials for the psychology of belief. The feeling one has 
in a novel or a play as to whether the behaviour of the characters 
is “right” is based upon unformulated knowledge of mental 
causality, and so is shrewdness in handling people. In such cases, 
the knowledge involved is pre-scientific, but it could not exist 
unless there were scientific laws which could be ascertained by 
sufficient study. 

There are a certain number of genuine causal laws of the 
kind in question, though so far they are mostly concerned with 
matters that have no great intrinsic interest. Take, for example, 
after-images: you look fixedly at a bright red object, and then 
shut your eyes; you see first a gradually fading red image, and 
then a green image, of approximately the same shape. This is 
a law for which the evidence is purely introspective. Or again, 
take a well-known illusion : 

In the figure the two horizontal lines are parallel, but they look 
as if they approached each other towards the right. This again 
is a law for which the evidence is purely introspective. In both 
cases there are physiological explanations, but they do not 
invalidate the purely psychological laws. 



I conclude that, while some psychological laws involve 
physiology, others do not. Psychology is a science distinct from 
physics and physiology, and in part independent of them. All 
the data of physics are also data of psychology, but not vice 
versa ; data belonging to both are made the basis of quite different 
inferences in the two sciences. Introspection is valid as a source 
of data, and is to a considerable extent amenable to scientific 

There is much in psychology that is genuinely scientific 
although it lacks quantitative precision. Take, for example, the 
analysis of our spatial perceptions, and the building up of the 
common-sense notion of space from its sensational foundations. 
Berkeley’s theory of vision, according to which everything looks 
flat, is disproved by the stereoscope. The process by which we 
learn in infancy to touch a place that we see can be studied by 
observation. So can volitional control: a baby a few months old 
can be watched learning with delight to move its toes at will, 
instead of having to look on passively while they wriggle in 
purely reflex movements. When, in later life, you acquire some 
skill, such as riding a bicycle, you find yourself passing through 
stages: at first you will certain movements of your own body, 
in the hope that they will cause the desired movements of the 
bicycle, but afterwards you will the movements of the bicycle 
directly, and the necessary movements of your body result auto- 
matically. Such experiences throw much light on the psychology 
of volition. 

Much psychology is involved in connecting sensory stimuli 
with the beliefs to which they give rise. I am thinking of such 
elementary occurrences as thinking “there’s a cat” when certain 
coloured patches in motion pass across your field of vision. It 
is obvious that the same sensory stimulus could be caused other- 
wise than by a cat, and your belief would then be false. You 
may see a room reflected in a mirror, and think that it is “real”. 
By studying such occurrences we become aware that a very 
large part of what we think we perceive consists of habits caused 
by past experience. Our life is full of expectations of which, 
as a rule, we only become aware when they are disappointed. 
Suppose you see half of a horse that is just coming round a 
corner; you may be very little interested, but if the other half 
proved to be cow and not horse you would experience a shock 

65 E 

HUMAN knowledge: its scope and limits 

of surprise which would be almost unendurable. Yet it must be 
admitted that such an occurrence is logically possible. 

The connection of pleasure and pain and desire with habit- 
formation can be studied experimentally. Pavlov, whose work 
nowhere appeals to introspection, put a dog in front of two doors, 
on one of which he had drawn an ellipse and on the other a 
circle. If the dog chose the right door he got his dinner; if he 
chose the wrong one he got an electric shock. Thus stimulated, 
the dog’s progress in geometry was amazingly rapid. Pavlov 
gradually made the ellipse less and less eccentric, but the dog 
still distinguished correctly, until the ratio of minor to major 
axis was reduced to 8:9, when the poor beast had a nervous 
breakdown. The utility of this experiment in connection with 
schoolboys and criminals is obvious. 

Or, again, take the question: why do we believe what we do? 
In former times, philosophers would have said it was because 
God had implanted in us a natural light by which we knew the 
truth. In the early nineteenth century they might have said it 
was because we had weighed the evidence and found a preponder- 
ance on one side. But if you ask a modern advertiser or political 
propagandist he will give you a more scientific and more 
depressing answer. A large proportion of our beliefs are based 
on habit, conceit, self-interest, or frequent iteration. The advertiser 
relies mainly on the last of these, but if he is clever he combines 
it skilfully with the other three. It is hoped that by studying 
the psychology of belief, those who control propaganda will in 
time be able to make anybody believe anything. Then the 
totalitarian State will become invincible. 

In regard to human knowledge there are two questions that 
may be asked: first, what do we know? and second, how do we 
know it? The first of these questions is answered by science, 
which tries to be as impersonal and as dehumanized as possible. 
In the resulting survey of the universe it is natural to start with 
astronomy and physics, which deal with what is large and what 
is universal; life and mind, which are rare and have, apparently, 
little influence on the course of events, must occupy a minor 
position m this impartial survey. But in relation to our second 
question, namely, how do we come by our knowledge, psychology 
is the most important of the sciences. Not only is it necessary 
to study psychologically the processes by which we draw inferences, 



but it turns out that all the data upon which our inferences 
should be based are psychological in character, that is to say, 
they are experiences of single individuals. The apparent publicity 
of our world is in part delusive and in part inferential; all the 
raw material of our knowledge consists of mental events in the 
lives of separate people. In this region, therefore, psychology is 




Chapter I 


L anguage, like other things of mysterious importance, such 
as breath, blood, sex, and lightning, has been viewed 
superstitiously ever since men were capable of recording 
their thoughts. Savages fear to disclose their true name to an 
enemy, lest he should work evil magic by means of it. Origen 
assures us that pagan sorcerers could achieve more by using 
the sacred name Jehovah than by means of the names Zeus, 
Osiris, or Brahma. Familiarity makes us blind to the linguistic 
emphasis in the Commandment: “Thou shalt not take the name 
of the Lord in vain.” The habit of viewing language superstitiously 
is not yet extinct. “In the beginning was the Word”, says our 
version of St. John’s Gospel, and in reading some logical 
positivists I am tempted to think that their view is represented 
by this mistranslated text. 

Philosophers, being bookish and theoretical folk, have been 
interested in language chiefly as a means of making statements 
and conveying information, but this is only one of its purposes, 
and perhaps not the most primitive. What is the purpose of 
language to a sergeant-major? On the one hand there is the 
language of words of command, designed to cause identical 
simultaneous bodily movements in a number of hearers ; on the 
other hand there is bad language, designed to cause humility 
in those in whom the expected bodily movements have not been 
caused. In neither case are words used, except incidentally, to 
state facts or convey information. 

Language can be used to express emotions, or to influence 
the behaviour of others. Each of these functions can be performed, 
though with less adequacy, by pre-linguistic methods. Animals 
emit shrieks of pain, and infants, before they can speak, can express 
rage, discomfort, desire, delight, and a whole gamut of feelings, 
by cries and gurgles of different kinds. A sheep dog emits imper- 
atives to his flock bv means hardly distinguishable from those 
that the shepherd employs towards him. Between such noises 
and speech no sharp line can be drawn. When the dentist hurts 
you, you may emit an involuntary groan; this does not count as 


human knowledge: its scope AND LIMITS 

speech. But if he says “let me know if I hurt you”, and you 
then make the very same sound, it has become speech, and 
moreover speech of the sort intended to convey information. 
This example illustrates the fact that, in the matter of language 
as in other respects, there is a continuous gradation from animal 
behaviour to that of the most precise man of science, and from 
pre-linguistic noises to the polished diction of the lexicographer. 

A sound expressive of emotion I shall call an “interjetion”. 
Imperatives and interjections can already be distinguished in 
the noises emitted by animals. When a hen clucks at her brood 
of chickens, she is uttering imperatives, but when she squawks 
in terror she is expressing emotion. But as appears from your 
groan at the dentist s, an interjection may convey information, 
and the outside observer cannot tell whether or not it is intended 
to do so. Gregarious animals emit distinctive noises when they 
find food, and other members of the herd are attracted when 
they hear these noises, but we cannot know whether the noises 
merely express pleasure or are also intended to state “food here”. 

Whenever an animal is so constructed that a certain kind of 
circumstance causes a certain kind of emotion, and a certain 
kind of emotion causes a certain kind of noise, the noise conveys 
to a suitable observer two pieces of information, first, that the 
animal has a certain kind of feeling, and second, that a certain 
kind of circumstance is present. The sound that the animal 
emits is public, and the circumstance may be public — e.g. the 
presence of a shoal of fish if the animal is a sea-gull. The animal’s 
cry may act directly on the other members of its species, and we 
shall then say that they “understand” its cry. But this is to suppose 
a “mental” intermediary between the hearing of the cry and the 
bodily reaction to the sound, and there is no real reason to suppose 
any such intermediary except when the response is delayed. 
Much of the importance of language is connected with delayed 
responses, but I will not yet deal with this topic. 

Language has two primary purposes, expression and com- 
munication. In its most primitive forms it differs little from some 
other forms of behaviour. A man may express sorrow by sighing, 
or by saying “alas!” or “woe is me!” He may communicate by 
pointing or by saying “look”. Expression and communication 
are not necessarily separated; if you say “look” because you see 
a ghost, you may say it in a tone that expresses horror. This 

7 * 


applies not only to elementary forms of language; in poetry, and 
especially in songs, emotion and information are conveyed by 
the same means. Music may be considered as a form of language 
in which emotion is divorced from information, while the telephone 
book gives information without emotion. But in ordinary speech 
both elements are usually present. 

Communication does not consist only of giving information; 
commands and questions must be included. Sometimes the two 
are scarcely separable: if you are walking with a child, and you 
say “there’s a puddle there”, the command “don’t step in it” 
is implicit. Giving information may be due solely to the fact that 
the information interests you, or may be designed to influence 
behaviour. If you have just seen a street accident, you will wish 
to tell your friends about it because your mind is full of it ; but 
if you tell a child that six times seven is forty-two you do so 
merely in the hope of influencing his (verbal) behaviour. 

Language has two interconnected merits: first, that it is social, 
and second that it supplies public expression for “thoughts” 
which would otherwise remain private. Without language, or 
some pre-linguistic analogue, our knowledge of the environment 
is confined to what our own senses have shown us, together with 
such inferences as our congenital constitution may prompt; 
but by the help of speech we are able to know what others can 
relate, and to relate what is no longer sensibly present but only 
remembered. When we see or hear something which a companion 
is not seeing or hearing, we can often make him aware of it by the 
one word “look” or “listen”, or even by gestures. But if half an 
hour ago we saw a fox, it is not possible to make another person 
aware of this fact without language. This depends upon the 
fact that the word “fox” applies equally to a fox seen or a fox 
remembered, so that our memories, which in themselves are 
private, are represented to others by uttered sounds, which are 
public. Without language, only that part of our life which consists 
of public sensations would be communicable, and that only to 
those so situated as to be able to share the sensations in question. 

It will be seen that the utility of language depends upon the 
distinction between public and private experiences, which is 
important in considering the empirical basis of physics. This 
distinction, in turn, depends partly on physiology, partly on the 
persistence of sound-waves and light quanta, which makes 


HUMAN knowledge: its scope and limits 

possible the two forms of language, speech and writing. Thus 
language depends upon physics, and could not exist without the 
approximately separable causal chains which, as we shall see, 
make physical knowledge possible, and since the publicity of 
sensible objects is only approximate, language applying to them, 
considered socially, must have a certain lack of precision. I need 
hardly say that I am not asserting that the existence of language 
requires a knowledge of physics. What I am saying is that language 
would be impossible if the physical world did not in fact have 
certain characteristics, and that the theory of language is at certain 
points dependent upon a knowledge of the physical world. 
Language is a means of externalizing and publicizing our own 
experiences. A dog cannot relate his autobiography; however 
eloquently he may bark, he cannot tell you that his parents were 
honest though poor. A man can do this, and he does it by 
correlating “thoughts” with public sensations. 

Language serves not only to express thoughts, but to make 
possible thoughts which could not exist without it. It is sometimes 
maintained that there can be no thought without language, 
but to this view I cannot assent : I hold that there can be thought, 
and even true and false belief, without language. But however 
that may be, it cannot be denied that all fairly elaborate thoughts 
require words. I can know, in a sense, that I have five fingers, 
without knowing the word “five”, but I cannot know that the 
population of London is about eight millions unless I have 
acquired the language of arithmetic, nor can I have any thought 
at all closely corresponding to what is asserted in the sentence : “the 
ratio of the circumference of a circle to the diameter is approxi- 
mately 3-14159”. Language, once evolved, acquires a kind of 
autonomy : we can know, especially in mathematics, that a sentence 
asserts something true, although what it asserts is too complex 
to be apprehended even by the best minds. Let us consider for 
a moment what happens psychologically in such cases. 

In mathematics, we start from rather simple sentences which 
we believe ourselves capable of understanding, and proceed, by 
rules of inference which we also believe ourselves to understand, 
to build up more and more complicated symbolic statements, 
which, if our initial assumptions are true, must be true whatever 
they may mean. As a rule it is unnecessary to know what they 
“mean”, if their “meaning” is taken to be a thought which 



might occur in the mind of a superhuman mathematical genius. 
But there is another kind of “meaning”, which gives occasion 
for pragmatism and instrumentalism. According to those who 
adopt this view of “meaning”, what a complicated mathematical 
sentence does is to give a rule for practical procedure in certain 
kinds of cases. Take, for instance, the above statement about the 
ratio of the circumference of a circle to the diameter. Suppose 
you are a brewer, and you desire hoops of a given diameter for 
your beer barrels, then the sentence gives you a rule by which 
you can find out how much material you will need. This rule 
may consist of a fresh sentence for each decimal point, and there 
is therefore no need ever to grasp its significance as a whole. 
The autonomy of language enables you to forego this tedious 
process of interpretation except at crucial moments. 

There are two other uses of language that are of great import- 
ance; it enables us to conduct our transactions with the outer 
world by means of symbols that have (l) a certain degree of 
permanence in time, (2) a considerable degree of discreteness in 
space. Each of these merits is more marked in writing than in 
speech, but is by no means wholly absent in speech. Suppose 
you have a friend called Mr. Jones. As a physical object his 
boundaries are somewhat vague, both because he is continually 
losing and acquiring electrons, and because an electron, being a 
distribution of energy, does not cease abruptly at a certain distance 
from its centre. The surface of Mr. Jones, therefore, has a certain 
ghostly impalpable quality, which you do not like to associate 
with your solid-seeming friend. It is not necessary to go into the 
niceties of theoretical physics in order to show that Mr. Jones is 
sadly indeterminate. When he is cutting his toe nails, there is 
a finite time, though a short one, during which it is doubtful 
whether the parings are still part of him or not. When he eats 
a mutton chop, at what moment does it become part of him? 
When he breathes out carbon dioxide, is the carbon part of him 
until it passes his nostrils ? Even if we answer in the affirmative, 
there is a finite time during which it is questionable whether 
certain molecules have or have not passed beyond his nostrils. 
In these and other ways, it is doubtful what is part of Mr. Jones 
and what is not. So much for spatial vagueness. 

There is the same problem as regards time. To the question 
“what are you looking at ?” you may answer “Mr. Jones”, although 


HUMAN knowledge: its scope and limits 

at one time you see him full-face, at another in profile, and at 
another from behind, and although at one time he may be running 
a race and at another time dozing in an arm-chair. There is 
another question, namely “what are you thinking of?” to which 
you may also answer “Mr. Jones”, though what is actually in 
your mind may be very different on different occasions : it may be 
Mr. Jones as a baby, or Mr. Jones being cross because his break- 
fast is late, or Mr. Jones receiving the news that he is to be 
knighted. What you are experiencing is very different on these 
various occasions, but for many practical purposes it is convenient 
to regard them as all having a common object, which we suppose 
to be the meaning of the name “Mr. Jones”. This name, especially 
when printed, though it cannot wholly escape the indefiniteness 
and transience of all physical objects, has much less of both than 
Mr. Jones has. Two instances of the printed words “Mr. Jones” 
are much more alike than (for instance) the spectacle of Mr. 
Jones running and the memory of Mr. Jones as a baby. And each 
instance, if printed, changes much more slowly than Mr. Jones 
does: it does not eat or breathe or cut its toe nails. The name, 
accordingly, makes it much easier than it would otherwise be to 
think of Mr. Jones as a single quasi-permanent entity, which, 
though untrue, is convenient in daily life. 

Language, as appears from the above discussion of Mr. Jones, 
though a useful and even indispensable tool, is a dangerous one, 
since it begins by suggesting a definiteness, discreteness, and 
quasi-permanence in objects which physics seems to show that 
they do not possess. The philosopher, therefore, is faced with the 
difficult task of using language to undo the false beliefs that it 
suggests. Some philosophers, who shrink from the problems and 
uncertainties and complications involved in such a task, prefer 
to treat language as autonomous, and try to forget that it is 
intended to have a relation to fact and to facilitate dealings with 
the environment. Up to a point, such a treatment has great 
advantages: logic and mathematics would not have prospered as 
they have done if logicians and mathematicians had continually 
remembered that symbols should mean something. “Art for 
art's sake” is a maxim which has a legitimate sphere in logic 
as in painting (though in neither case does it give the whole 
truth). It may be that singing began as an incident in courtship, 
and that its biological purpose was to promote sexual intercourse ; 



but this fact (if it be a fact) will not help a composer to produce 
good music. Language is useful when you wish to order a meal 
in a restaurant, but this fact, similarly, is of no importance to the 
pure mathematician. 

The philosopher, however, must pursue truth even at the 
expense of beauty, and in studying language he must not let 
himself be seduced by the siren songs of mathematics. Language, 
in its beginnings, is pedestrian and practical, using rough and 
ready approximations which have at first no beauty and only a 
very limited degree of truth. Subsequent refinements have too 
often had aesthetic rather than scientific motives, but from the 
inquiry upon which we are about to embark aesthetic motives 
must, however reluctantly, be relentlessly banished. 


Chapter II 


41 stensive definition” may be defined as “any process 

I 1 by which a person is taught to understand a word other- 
wise than by the use of other words”. Suppose that, 
knowing no French, you are shipwrecked on the coast of 
Normandy : you make your way into a farmhouse, you see bread 
on the table, and, being famished, you point at it with an 
inquiring gesture. If the farmer thereupon says “pain 9 \ you will 
conclude, at least provisionally, that this is the French for 
“bread”, and you will be confirmed in this view if the word is 
not repeated when you point at other kinds of eatables. You 
will then have learnt the meaning of the word by ostensive 
definition. It is clear that, if you know no French and your teacher 
knows no English, you must depend upon this process during 
your first lessons, since you have no linguistic means of com- 

The process of ostensive definition, however, is better exem- 
plified when the learner knows no language at all than when 
he already possesses a language of his own. An adult knows that 
there are words, and will naturally suppose that the French 
have a way of naming bread. His knowledge takes the form: 
“ 4 Pain 9 means ‘bread’ ”. It is true that, when you were ship- 
wrecked, it was by means of actual bread that you acquired this 
knowledge, but if you had been shipwrecked with a dictionary 
the actual bread would not have been necessary. There are two 
stages in the acquisition of a foreign language, the first that in 
which you only understand by translating, the second that in 
which you can “think” in the foreign language. In the first stage 
you know that “pain” means “bread”, in the second stage you 
know that it means bread. The infant, possessing as yet no 
language, has to begin with the second stage. His success does 
credit to the capacities of the infant mind. 

Knowing a language has two aspects, passive and active: 
passive when you understand what you hear, active when you 
can speak yourself. Dogs to some degree achieve the former, 
and children usually achieve it some time before the latter. 



Knowing a language does not mean a capacity for explicit explana- 
tion of what its words signify; it means that hearing the words 
has appropriate effects, and using them has appropriate causes. 
I have sometimes, in the course of travel, watched a quarrel 
springing up between two men whose language I did not under- 
stand, and it was difficult not to feel their mounting excitement 
ridiculous. But probably the first was accusing the second of 
being the offspring of parents who were not married, and the 
second was retorting that the first’s wife was unfaithful. If I had 
understood, the effect of the insult and the cause of the retort 
would have been obvious. As this example illustrates, a person 
knows a language when hearing certain sounds has certain effects 
and uttering them has certain causes. The process by which, 
in the infant, the establishment of these causal laws is begun, 
is the process of ostensive definition. 

Ostensive definition, in its earliest form, requires certain 
conditions. There must be a feature of the environment which is 
noticeable, distinctive, emotionally interesting, and (as a rule) 
frequently recurring, and the adult must frequently utter the 
name of this feature at a moment when the infant is attending 
to it. Of course there are risks of error. Suppose the child has 
milk in a bottle. You may each time say “milk” or each time say 
“bottle”. In the former case the child may think “milk” is the 
right word for a bottle of water; in the latter case, he may think 
“bottle” the right word for a glass of milk. To avoid such errors, 
you should in theory apply Mill’s inductive canons, remembering 
that induction is a bodily habit, and only by courtesy a logical 
process. Instead of saying merely “milk” or merely “bottle”, 
you should say “bottle of milk”; you should then, on appropriate 
occasions, say “glass of milk” and “bottle of water”. In time, 
by the use of Mill’s canons, the infant, if he survives, will learn 
to speak correctly. But I am not giving practical pedagogic 
advice; I am merely exemplifying a theory. 

The passive part in ostensive definition is merely the familiar 
business of association or the conditioned reflex. If a certain 
stimulus A produces in a child a certain reaction R, and is fre- 
quently experienced in conjunction with the word B, it will happen 
in time that B will produce the reaction R, or some part of it. 
As soon as this has happened, the word B has acquired a “mean- 
ing” for the child: it “means” A. The meaning may not be quite 


human knowledge: its scope and limits 

what the adult intended: the adult may have intended “bottle” 
and the child may understand the word as meaning milk. But 
that does not prevent the child from possessing a word that has 
meaning; it only signifies that the child's language is not yet 
correct English. 

When an experience causes violent emotion, repetition may be 
unnecessary. If a child, after learning to understand “milk”, 
is given milk so hot as to scald his mouth, and you say “hot”, 
he may ever after understand this word. But when an experience 
is uninteresting, many repetitions may be necessary. 

The active part in the learning of language requires other 
capacities, which however, are of less philosophic interest. Dogs 
cannot learn human speech because they are anatomically in- 
capable of producing the right sounds. Parrots, though they can 
produce more or less the right sounds, seem incapable of acquiring 
the right associations, so that their words do not have meaning. 
Infants, in common with the young of the higher animals, have 
an impulse to imitate adults of their own species, and therefore 
try to make the sounds that they hear. They may, on occasion, 
repeat sounds like a parrot, and only subsequently discover the 
“meaning” of the sounds. In that case the sounds cannot count 
as words until they have acquired meaning for the child. For 
every child it is a discovery that there are words, i.e. sounds 
with meaning. Learning to utter words is a joy to the child, largely 
because it enables him to communicate his wishes more definitely 
than he had been able to do by crying and making gestures. It 
is owing to this pleasure that children go through the mental 
labour and muscular practice involved in learning to talk. 

In general, though not universally, repetition is necessary for 
an ostensive definition, for ostensive definition consists in the 
creation of a habit, and habits, as a rule, are learned gradually. 
The exceptional cases are illustrated by the proverbs “once 
bit, twice shy” and “the burnt dblild dreads the fire”. Apart from 
such unusually emotional matters, the words that have ostensive 
definitions denote frequently recurring features of the environ- 
ment, such as the members of the family, foods, toys, pet animals, 
etc. This involves the process of recognition, or something of 
the kind. Although a child’s mother looks somewhat different 
on different occasions, he thinks of her (when he begins to think) 
as always the same person, and feels no difficulty in applying the 



same name to her various epiphanies. Language, from the start, 
or rather from the start of reflection on language, embodies the 
belief in more or less permanent persons and things. This is 
perhaps the chief reason for the difficulty of any philosophy 
which dispenses with the notion of substance. If you were to 
tell a child that his mother is a series of sensible impressions, 
connected by similarity and causal relations, but without 
material identity, and if by a miracle you could make him 
understand what you meant, he would consider you demented 
and be filled with indignation. The process called “recognition” 
is therefore one that demands investigation. 

Recognition, as a physiological or psychological occurrence, 
may or may not be veridical. It fails in an every-day sense to be 
veridical when we mistake one of two twins for the other, but 
it may be metaphysically misleading even when it is correct 
from the standpoint of common sense. Whether there is anything 
identical, and if so what, between two different appearances of 
Mr. A, is a dark and difficult question, which I shall consider 
in connection with proper names. For the moment I wish to 
consider recognition as a process which actually occurs, without 
regard to its interpretation. 

The first stage in the development of this process is repetition 
of a learnt reaction when the stimulus is repeated. It must be a 
learnt reaction, since recognition must grow out of a process 
involving something, in later reactions to a given stimulus, which 
was not present in the first reaction. Suppose, for instance, you 
give a child a glass of milk containing bitter medicine: the first 
time he drinks the doctored milk and makes a face, but the 
second time he refuses the milk. This is subjectively something 
like recognition, even if the second time he is mistaken in 
supposing that the milk contains medicine. It is clear that this 
process may be purely physiological, and that it involves only 
similarity, not identity, in stimulus and response. The learning 
of words by ostensive definition can be brought wholly within 
this primitive stage. The child’s world contains a number of 
similar stimuli to which he has learnt to respond by similar 
noises, namely those that are instances of the word “milk”; 
it contains also another set of similar stimuli to which he has 
learnt to respond by instances of the word “mother”. In this 
there is nothing involving any beliefs or emotions in the child. 



HUMAN knowledge: its scope and limits 

It is only as a result of subsequent reflection that the child, now 
become a philosopher, concludes that there is one word, “mother”, 
and one person, Mother. I believe this first step in philosophy 
to be mistaken. The word “mother”, I should say, is not a single 
entity, but a class of similar noises; and Mother herself is also 
not a single entity, but a class of causally connected occurrences. 
These speculations, however, are irrelevant to the process of 
ostensive definition, which, as we have just seen, requires only 
the very first stage on the road towards what would usually 
count as recognition, namely similar learnt responses to similar 

This primitive form of recognition is relevant in the analysis 
of memory and in explaining the similarity of an idea to an 
impression (to borrow Hume’s phraseology). When I remember 
a past event, I cannot make it itself occur again, though I may 
be able to make a similar event occur. But how do I know that 
the new event is similar to the old one ? Subjectively, I can only 
know by comparing an idea with an impression: I have an idea 
of the past event and an impression of the present event, and 
I perceive that they are similar. But this is not sufficient, since 
it does not prove that my idea of the past event is similar to my 
impression of th t past event when it existed. This, in fact, cannot 
be proved, and is in some sense one of the premisses of knowledge. 
But although it cannot be strictly proved, it can be in various 
ways confirmed. You may describe Mr. A while he is present, 
and your description may be recorded on a dictaphone. You 
may later describe him from memory, and compare your new 
description with the dictaphone record. If they agree closely, 
your memory may be accepted as correct. 

This illustration depends upon a fact which is fundamental 
in this subject, namely, that we apply the same words to ideas 
as to the impressions which are their prototypes. This explains 
the possibility of learning a word ostensively by means of a single 
sensible occurrence. I saw Disraeli once, and once only, and was 
told, at the moment, “that’s Dizzy”. I have since very frequently 
remembered the occurrence, with the name “Dizzy” as an 
essential part of the memory. This has made it possible for a 
habit to be formed by repetition of the idea (in Hume’s sense), 
although the impression has never been repeated. It is obvious 
that ideas differ from impressions in various ways, but their 



similarity to their prototypes is vouched for by the fact that 
they cause the same words. The two questions, “what are you 
looking at?” and “what are you thinking of?” may, on two 
different occasions, be answered identically. 

Let us consider the different kinds of words that are commonly 
learnt by means of ostensive definitions. What I have in mind is 
a logical form of the grammatical doctrine of parts of speech. 

We have already had occasion for a preliminary consideration 
of proper names. I shall say no more about them at present, as 
they will be the subject of a separate chapter. 

Next come names of species: man, woman, cat, dog, etc. A 
species of this sort consists of a number of separate individuals, 
having some recognizable degree of likeness to each other. In 
biology before Darwin, “species” was a prominent concept. 
God had created a pair of each species, and different species 
could not interbreed, or, in the exceptional cases when they could, 
such as horse and ass, the offspring was sterile. There was an 
elaborate hierarchy of genera, families, orders, etc. This kind 
of classification, which was and is convenient in biology, was 
extended by the scholastics to other regions, and impeded logic 
by creating the notion that some ways of classifying are more 
correct than others. As regards ostensive definition, different 
experiences will produce different results. Most children learn 
the word “dog” ostensively; some learn in this way the kinds of 
dogs, collies, St. Bernards, spaniels, poodles, etc., while others, 
who have little to do with dogs, may first meet with these words 
in books. No child learns the word “quadruped” ostensively, 
still less the word “animal” in the sense in which it includes 
oysters and limpets. He probably learns “ant”, “bee”, and 
“beetle” ostensively, and perhaps “insect”, but if so he will 
mistakenly include spiders until corrected. 

(^Names of substances not obviously collections of individuals, 
such as “milk”, “bread”, “wood”, are apt to be learnt ostensively 
when they denote things familiar in every-day life. The atomic 
theory is an attempt to identify this class of objects with the 
former, so that milk, for instance, is a collection of milky 
individuals (molecules), just as the human race is a collection of 
men, women, and children. J But to unscientific apprehension 
such names of substances are not to be assimilated to species 
composed of separate individuals. 


HUMAN knowledge: its scope and limits 

Next come qualities: red, blue, hard, soft, hot, cold, etc. 
Many of these are usually learnt ostensively, but the less common 
ones, such as vermilion, may be described by their similarities 
and differences. 

Names of certain relations, such as “up”, “down”, “right”, 
“left”, “before”, “after”, are usually learnt ostensively. So are 
such words as “quick” and “slow”. 

There are a number of words of the sort that I call “ego- 
centric”, which differ in meaning according to the speaker and 
his position in time and space. Among these the simple ones are 
learnt ostensively, for instance “I”, “you”, “here”, “now”. 
These words raise problems which we will consider in a later 

All the words I have mentioned hitherto belong to the public 
world. A spectator can see when a certain feature of the public 
environment is attracting a child’s attention, and then mention 
the name of this feature. But how about private experiences, 
such as stomach-ache, pain, or memory? Certainly some words 
denoting private kinds of experience are learnt ostensively. This 
is because the child shows in behaviour what he is feeling : there 
is a correlation between e.g. pain and tears. 

There are no definite limits to what can be learnt by ostensive 
definition. “Cross”, “crescent”, “swastika” can be learnt in this 
way, but not “chiliagon”. But the point where this method of 
learning becomes impossible depends upon the child’s experience 
and capacity. 

The words so far mentioned are all capable of being used as 
complete sentences, and are in fact so used in their most primitive 
employment. “Mother”, “dog”, “cat”, “milk”, and so on, may 
be used alone to express either recognition or desire. “Hard”, 
“soft”, “hot”, “cold” would be more naturally used to express 
recognition than desire, and usually to express recognition 
accompanied by surprise. If the toast is uneatable because it is 
old you may say “hard”; if a ginger biscuit has lost its crispness 
by exposure to air you may say “soft”. If the bath scalds you, 
you say “hot”; if it freezes you, you say “cold”. “Quick” is 
frequently used by parents as an imperative; “slow” is used 
similarly on roads and railways where there is a curve. The words 
“up” and “down” are habitually used as complete sentences 
by lift-boys; “in” and “out” are similarly used at turnstiles. 



“Before” and “after” are used as complete sentences in advertise- 
ments of hair-restorers. And so on and so on. It is to be noted 
that not only substantives and adjectives, but adverbs and pre- 
positions, may on occasion be used as complete sentences. 

I think the elementary uses of a word may be distinguished as 
indicative, imperative, and interrogative. When a child sees his 
mother coming, he may say “mother!”; this is the indicative use. 
When he wants her, he calls “mother!”; this is the imperative 
use. When she dresses up as a witch and he begins to pierce the 
disguise, he may say “mother?”; this is the interrogative use. 
The indicative use must come first in the acquisition of language, 
since the association of word and object signified can only be 
created by the simultaneous presence of both. But the imperative 
use very quickly follows. This is relevant in considering what 
we mean by “thinking of” an object. It is obvious that the child 
who has just learnt to call his mother has found verbal expression 
for a state in which he had often been previously, that this state 
was associated with his mother, and that it has now become 
associated with the word “mother”. Before language, his state 
was only partially communicable; an adult, hearing him cry, 
could know that he wanted something, but had to guess what it 
was. But the fact that the word “mother!” expresses his state 
shows that, even before the acquisition of language, his state 
had a relation to his mother, namely the relation called “thinking 
of”. This relation is not created by language, but antedates it. 
What language does is to make it communicable. 

“Meaning” is a word which must be interpreted somewhat 
differently according as it is applied to the indicative or the 
imperative. In the indicative, a word A means a feature B of the 
environment if, (i) when B is emphatically present to attention, 
A is uttered, or there is an impulse to utter A, and (2) when A 
is heard it arouses what may be called the “idea” of B, which 
shows itself either in looking for B or in behaviour such as would 
be caused by the presence of B. Thus in the indicative a word 
“means” an object if the sensible presence of the object causes 
the utterance of the word, and the hearing of the word has effects 
analogous, in certain respects, to the sensible presence of the 

The imperative use of a word must be distinguished according 
as it is heard or uttered. Broadly speaking, an imperative heard 


human knowledge: its scope and limits 

— e.g. the word of command in the army — is understood when 
it causes a certain kind of bodily movement, or an impulse towards 
such a movement. An imperative uttered expresses a desire, and 
therefore requires the existence of an “idea” of the intended 
effect. Thus while it “expresses” something in the speaker, it 
“means” the external effect which it commands. The distinction 
between what is “meant” and what is “expressed” is essential 
in this use of words. 

We have been concerned, in this chapter, only with the most 
primitive uses of the most primitive words. We have not considered 
the use of words in narrative or in hypothesis or in fiction, nor 
have we examined logical words such as “not”, “or”, “all”, and 
“some”; we have not inquired how learners acquire the correct 
use of such words as “than” or “of”, which do not denote 
recognizable features of any sensible environment. What we 
have decided is that a word may become associated with some 
notable feature of the environment (in general, one that occurs 
frequently), and that, when it is so associated, it is also associated 
with something that may be called the “idea” or “thought” of 
this feature. When such an association exists, the word “means” 
this feature of the environment; its utterance can be caused by 
the feature in question, and the hearing of it can cause the “idea” 
of this feature. This is the simplest kind of “meaning”, out of 
which other kinds are developed. 


Chapter III 

T here is a traditional distinction between “proper” names 
and “class” names, which is explained as consisting in 
the fact that a proper name applies, essentially, to only 
one object, whereas a class name applies to all objects of a certain 
kind, however numerous they may be. Thus “Napoleon” is a 
proper name, while “man” is a class name. It will be observed 
that a proper name is meaningless unless there is an object of 
which it is the name, but a class name is not subject to any such 
limitation. “Men whose heads do grow beneath their shoulders” 
is a perfectly good class name, although there are no instances 
of it. Again, it may happen that there is only one instance of a 
class name, e.g. “satellite of the earth”. In such a case, the one 
member may have a proper name (“the moon”), but the proper 
name does not have the same meaning as the class name, and has 
different syntactical functions. E.g. we can say: “ ‘Satellite of the 
earth’ is a unit class”, but we cannot say “the moon is a unit 
class”, because it is not a class, or at any rate not a class of the 
same logical type as “satellite of the earth”, and if taken as a 
class (e.g. of molecules) it is many, not one. 

Many difficult questions arise in connection with proper names 
Of these there are two that are especially important: first, what 
is the precise definition of proper names? second, is it possible 
to express all our empirical knowledge in a language containing 
no proper names? This second question, we shall find, takes us 
to the heart of some of the most ancient and stubborn of philo- 
sophical disputes. 

In seeking a definition of “proper name”, we may approach 
the subject from the point of view of metaphysics, logic, physics, 
syntax, or theory of knowledge. I will say a few preliminary words 
about each of these. 

A. Metaphysical — It is fairly obvious that proper names owe 
their existence in ordinary language to the concept of “substance” 
— originally in the elementary form of “persons” and “things”. 
A substance or entity is named, and then properties are assigned 
to it. So long as this metaphysic was accepted, there was no 


human knowledge: its scope and limits 

difficulty as to proper names, which were the designations of such 
substances as were sufficiently interesting. Sometimes, it is true, 
we should give a name to a collection of substances, such as 
France or the sun. But such names, strictly speaking, were not 
necessary. In any case, we could extend our definition to embrace 
collections of substances. 

But most of us, nowadays, do not accept “substance” as a 
useful notion. Are we then to adopt, in philosophy, a language 
without proper names? Or are we to find a definition of “proper 
name” which does not depend on “substance”? Or are we to 
conclude that the conception of “substance” has been too hastily 
rejected? For the present, I merely raise these questions, without 
attempting to answer them. All that I want to make clear at the 
moment is that proper names, as ordinarily understood, are ghosts 
of substances. 

B. Syntactical . — It i9 clear that a syntactical definition of 
“proper name” must be relative to a given language or set of 
languages. In the languages of daily life, and also in most of those 
employed in logic, there is a distinction between subject and 
predicate, between relation-words and term-words. A “name” will 
be, in such languages, “a word which can never occur in a sentence 
except as a subject or a term-word”. Or again: a proper name 
is a word which may occur in any form of sentence not containing 
variables, whereas other words can only occur in sentences of 
appropriate form. Sometimes it is said that some words are 
“syncategorimatic”, which apparently means that they have no 
significance by themselves, but contribute to the significance of 
sentences in which they occur. According to this way of speaking, 
proper names are not syncategorimatic, but whether this can be 
a definition is a somewhat doubtful question. In any case, it is 
difficult to get a clear definition of the term “syncategorimatic”. 

The chief inadequacy of the above syntactical point of view is 
that it does not, in itself, help us to decide whether it is possible 
to construct languages with a different kind of syntax, in which 
the distinctions we have been considering would disappear. 

C. Logical . — Pure logic has no occasion for names, since its 
propositions contain only variables. But the logician may wonder, 
in his unprofessional moments, what constants could be sub- 
stituted for his variables. The logician announces, as one of his 
principles, that, if “fx” is true for every value of “#”, then "/»” 


is true, where “a” is any constant. This principle does not mention 
a constant, because “any constant” is a variable ; but it is intended 
to justify those who want to apply logic. Every application of logic 
or mathematics consists in the substitution of constants for 
variables; it is therefore essential, if logic or mathematics is to 
be applied, to know what sort of constants can be substituted 
for what sort of variables. If any kind of hierarchy is admitted 
among variables, “proper names” will be “constants which are 
values of variables of lowest type”. There are, however, a number 
of difficulties in such a view. I shall not therefore pursue it further. 

D. Physical . — There are here two points of view to be con- 
sidered. The first is that a proper name is a word designating any 
continuous portion of space-time which sufficiently interests us; 
the second is that, this being the function of proper names, they 
are unnecessary, since any portion of space-time can be described 
by its co-ordinates. Carnap {Logical Syntax , pp. 12-13) explains 
that latitude and longitude, or space- time co-ordinates, can be 
substituted for place-names. “The method of designation by 
proper names is the primitive one ; that of positional designation 
corresponds to a more advanced stage of science, and has con- 
siderable methodological advantages over the former.” In the 
language he employs, co-ordinates, he says, replace such words 
as “Napoleon” or “Vienna”. This point of view deserves full 
discussion, which I shall undertake shortly. 

E. Epistemological . — We have here, first, a distinction not 
identical with that between proper names and other words, but 
having perhaps some connection with it. This is the distinction 
between words having a verbal definition and words having only 
an ostensive definition. As to the latter, two points are obvious: 
(1) not all words can have verbal definitions; (2) it is largely 
arbitrary which words are to have only ostensive definitions. 
E.g. if “Napoleon” is defined ostensively, “Joseph Bonaparte” 
may be defined verbally as “Napoleon's oldest brother”. However, 
this arbitrariness is limited by the fact that, in the language of 
a given person, ostensive definitions are only possible within the 
limits of his experience. Napoleon’s friends might (subject to 
limitations) define him ostensively, but we cannot, since we can 
never say truly “ that is Napoleon”. There is obviously here a 
problem connected with that of proper names; how closely, I 
shall not discuss at present. 


HUMAN knowledge: its scope and limits 

We have, it is clear, a number of problems to consider, and, 
as is apt to happen in philosophy, it is difficult to be clear as to 
what precisely the problems are. I think we shall do best if we 
begin with Carnap’s substitution of co-ordinates for proper names. 
The question we have to consider is whether such a language 
can express the whole of our empirical knowledge. 

In Carnap’s system, a group of four numbers is substituted for 
a space-time point. He illustrates by the example “Blue x ly x 2 , 
# 3 , jc 4 , meaning “the position (x v x 2f x 3l x 4 ) is blue”, instead of 
“Blue (a)” meaning “the object a is blue”. But now consider 
such a sentence as “Napoleon was in Elba during part of 1814”. 
Carnap, I am sure, will agree that this sentence is true, and that 
its truth is empirical, not logical. But if we translate it into his 
language it will become a logical truth. “Napoleon” will be 
replaced by “all quartets of numbers enclosed within such-and- 
such boundaries”; so will “Elba”, and so will “1814”. We shall 
then be stating that these three classes of quartets have a common 
part. This, however, is a fact of logic. Clearly this is not what 
we meant. We give the name “Napoleon” to a certain region, not 
because we are concerned with topology, but because that region 
has certain characteristics which make it interesting. We may 
defend Carnap by supposing, adopting a schematic simplification, 
that “Napoleon” is to mean “all regions having a certain quality 
say N”, while “Elba” is to mean “all regions having the quality E”. 
Then “Napoleon spent some time in Elba” will become: “The 
regions having the quality N and those having the quality E 
overlap.” This is no longer a fact of logic. But it has interpreted 
the proper names of ordinary language as disguised predicates. 

But our schematic simplification is too violent. There is no 
quality, or collection of qualities, present wherever Napoleon was 
and absent wherever he was not. As an infant, he did not wear 
a cocked hat, or command armies, or fold his arms, while all these 
things were also done, at times, by other people. How, then, are 
we to define the word “Napoleon”? Let us continue to do our 
best for Carnap. In the moment of baptism, the priest decides 
that the name “Napoleon” is to apply to a certain small region 
in his neighbourhood, which has a more or less human shape, 
and that it is to apply to other future regions connected with this 
one, not only by continuity, which is not sufficient to secure 
material identity, but by certain causal laws, those, namely, which 



lead us to regard a body on two occasions as that of the same 
person. We may say: Given a temporally brief region having the 
characteristics of a living human body, it is an empirical fact that 
there are earlier and later regions connected with this one by 
physical laws, and having more or less similar characteristics ; the 
total of such regions is what we call a “person”, and one such 
region was called “Napoleon”. That naming is retro-active appears 
from a plaque on a certain house in Ajaccio saying: “Ici Napoleon 
fut con9u.” 

This may be accepted as an answer to the objection that, on 
Carnap’s view, “Napoleon was once in Elba” would be a pro- 
position of logic. It leaves, however, some very serious questions. 
We saw that “Napoleon” cannot be defined simply by qualities, 
unless we are to hold it impossible that there should be two 
exactly similar individuals. One of the uses of space-time, how- 
ever, is to differentiate similar individuals in different places. 
Carnap has his sentences “Blue (3)”, “Blue (4)”, etc., meaning 
“the place 3 is blue”, “the place 4 is blue”, etc. We can, it is 
supposed, distinguish blue in one place from blue in another. 
But how are the places distinguished? Carnap takes space-time 
for granted, and never discusses how space-time places are dif- 
ferentiated. In fact, in his system, space-time regions have the 
characteristics of substance. The homogeneity of space-time is 
assumed in physics, and yet it is also assumed that there are 
different regions, which can be distinguished. Unless we are to 
accept the objectionable metaphysics of substance, wc shall have 
to suppose the regions distinguished by differences of quality. 
We shall then find that the regions need no longer be regarded 
as substantial, but as bundles of qualities. 

Carnap’s co-ordinates, which replace names, are, of course, not 
assigned quite arbitrarily. The origin and the axes are arbitrary, 
but when they are fixed, the rest proceeds on a plan. The year 
which we call “1814” is differently named by the Mohammedans, 
who date from the Hejira, and by the Jewish era, which dates 
from the Creation. But the year we call “1815” will have the 
next number, in any system, to that given to what we call “1814”. 
It is because co-ordinates are not arbitrary that they are not 
names. Co-ordinates describe a point by its relations to the origin 
and the axes. But we must be able to say “ this is the origin”. If 
we are to be able to say this, we must be able to name the origin, 


HUMAN knowledge: its scope and limits 

or to describe it in some way, and at first sight it might be thought 
that any possible way would be found to involve names. Take, 
for instance, longitude. The origin of longitude is the meridian 
of Greenwich, but it might equally well be any other meridian. 
We cannot define “Greenwich” as “longitude o°, latitude 52 0 ”, 
because, if we do, there is no means of ascertaining where longi- 
tude o° is. When we say “longitude o° is the longitude of Green- 
wich”, what we say is satisfactory because we can go to Greenwich 
and say “this is Greenwich”. Similarly, if we live at (say) longi- 
tude 40° W., we can say “the longitude of this place is 40° W.”, 
and then we can define longitude o° by relation to this place. But 
unless we have a way of knowing some places otherwise than by 
latitude and longitude, latitude and longitude become unmeaning. 
When we ask “what are the latitude and longitude of New York?” 
we are not asking the same sort of question as we should be if 
we descended into New York by a parachute and asked “what 
is the name of this city?” We are asking: “How far is New York 
west of Greenwich and north of the equator?” This question 
supposes New York and Greenwich known and already named. 

It would be possible to assign a finite number of co-ordinates 
at haphazard, and then they would all be names. When (as is 
always done) they are assigned on a principle, they are descrip- 
tions, defining points by their relations to the origin and the axes. 
But these descriptions fail for the origin and the axes themselves, 
since, as regards them, the numbers are assigned arbitrarily. To 
answer the question “where is the origin?” we must have some 
method of identifying a place without mentioning its co-ordinates. 
It is the existence of such methods that is presupposed by the 
use of proper names. 

I conclude, for the moment, that we cannot wholly dispense 
with proper names by means of co-ordinates. We can perhaps 
reduce the number of proper names, but we cannot avoid them 
altogether. Without proper names we can express the whole of 
theoretical physics, but no part of history or geography; this, at 
least, is our provisional conclusion so far, but we shall find reason 
to modify it later. 

Let us consider a little further the substitution of descriptions 
for names. Somebody must be the tallest man now living in the 
United States. Let us suppose he is Mr. A. We may then, in 
place of “Mr. A”, substitute “the tallest man now living in the 



United States”, and this substitution will not, as a rule, alter the 
truth or falsehood of any sentence in which it is made. But it 
will alter the statement. One may know things about Mr. A that 
one does not know about the tallest man in the United States, 
and vice versa. One may know that Mr. A lives in Iowa, but not 
that the tallest man in the United States lives in Iowa. One may 
know that the tallest man in the United States is over ten years 
old, but one may not know whether Mr. A is man or boy. Then 
there is the proposition “Mr. A is the tallest man in the United 
States.” Mr. A may not know this; there may be a Mr. B who 
runs him close. But Mr. A certainly knows that Mr. A is Mr. A. 
This illustrates once more that there are some things which cannot 
be expressed by means of descriptions substituted for names. 

The names of persons have verbal definitions in terms of “this”. 
Suppose you are in Moscow and some one says “that’s Stalin”, 
then “Stalin” is defined as “the person whom you are now seeing” 
— or, more fully: “that series of occurrences, constituting a 
person, of which this is one”. Here “this” is undefined, but 
“Stalin” is defined. I think it will be found that every name 
applied to some portion of space-time can have a verbal definition 
in which the word “this”, or some equivalent, occurs. This, I 
should say, is what distinguishes the name of an historical character 
from that of an imaginary person, such as Hamlet. Let us take 
a person with whom we are not acquainted, say Socrates. We 
may define him as “the philosopher who drank the hemlock”, 
but such a definition does not assure us that Socrates existed, 
and if he did not exist, “Socrates” is not a name. What does 
assure us that Socrates existed ? A variety of sentences heard or 
read. Each of these is a sensible occurrence in our own experience. 
Suppose we find in the Encyclopaedia the statement “Socrates 
was an Athenian philosopher”. The sentence, while we see it, 
is a this , and our faith in the Encyclopaedia leads us to say “this 
is true”. We can define “Socrates” as “the person described in 
the Encyclopaedia under the name ‘Socrates’ ”. Here the name 
“Socrates” is experienced. We can of course define “Hamlet” in 
a similar way, but some of the propositions used in the definition 
will be false. E.g. if we say “Hamlet was a Prince of Denmark 
who was the hero of one of Shakespeare’s tragedies”, this is false. 
What is true is: “ ‘Hamlet’ is a word which Shakespeare pretends 
to be the name of a Prince of Denmark”. It would thus seem to 


HUMAN knowledge: its scope and limits 

follow that, apart from such words as “this” and “that”, every 
name is a description involving some this , and is only a name 
in virtue of the truth of some proposition. (The proposition may 
be only “this is a name”, which is false if this is “Hamlet”.) 

We must consider the question of minimum vocabularies. I call 
a vocabulary a “minimum” one if it contains no word which is 
capable of a verbal definition in terms of the other words of the 
vocabulary.^ Two minimum vocabularies dealing with the same 
subject-matter may not be equal ; there may be different methods 
of definition, some of which lead to a shorter residuum of 
undefined terms than others do. The question of minimum 
vocabularies is sometimes very important. Peano reduced the 
vocabulary of arithmetic to three words. It was an achievement in 
classical physics when all units were defined in terms of the units 
of mass, length, and time. The question I wish to discuss is: 
What characteristics must belong to a minimum vocabulary by 
means of which we can define all the words used in expressing 
our empirical knowledge or beliefs, in so far as such words have 
any precise meaning? More narrowly, to revert to a former 
example, what sort of minimum vocabulary is needed for 
“Napoleon was in Elba during part of 1814” and kindred state- 
ments? Perhaps, when we have answered this, we shall be able 
to define “names”. I assume, in the following discussion, that 
such historical-geographical statements are not analytic, that is 
to say, though they are true as a matter of fact, it would not be 
logically impossible for them to be false. 

Let us revert to the theory, which is suggested by what Carnap 
says, that “Napoleon” is to be defined as a certain region of 
space-time. We objected that, in that case, “Napoleon was for 
a time in Elba” is analytic. It may be retorted: yes, but to find 
out what is not analytic you must inquire why we give a name 
to the portion of space-time that was Napoleon. We do so because 
it had certain peculiar characteristics. It was a person, and when 
adult it wore a cocked hat. We shall then say: “This portion of 
space-time is a person, and in its later portions it wears a cocked 
hat; that portion of space- time is a small island; this and that 
have a common part”. We have here three statements, the first 
two empirical, the third analytic. This seems unobjectionable. 
It leaves us with the problem of assigning co-ordinates, and also 
with that of defining such terms as “person” and “island”. Such 



terms as “person” and “island” can obviously be defined in terms 
of qualities and relations; they are general terms, and not (one 
would say) such as lead to proper names. The assigning of co- 
ordinates requires the assigning of origin and axes. We may, for 
simplicity, ignore the axes and concentrate on the origin. Can 
the origin be defined ? 

Suppose, for example, you are engaged in planetary theory, not 
merely in a theoretical spirit, but with a view to the testing of 
your calculations by observations. Your origin, in that case, will 
have to be defined by something observable. It is universally 
agreed that absolute physical space-time is not observable. The 
things we can observe are, broadly speaking, qualities and spatio- 
temporal relations. We can say “I shall take the centre of the 
sun as my origin”. The centre of the sun is not observable, but 
the sun (in a sense) is. It is an empirical fact that I frequently 
have an experience which I call “seeing the sun”, and that I can 
observe what seem to be other people having a similar experience. 
“The sun” is a term which can be defined by qualities: round, 
hot, bright, of such-and-such apparent size, etc. It happens that 
there is only one object in my experience having these qualities, 
and that this object persists. I can give it a proper name, “the 
sun”, and say “I shall take the sun as my origin”. But since I 
have defined the sun by its qualities, it does not form part of 
a minimum vocabulary. It seems to follow that, while the words 
for qualities and spatio-temporal relations may form part of my 
minimum vocabulary, no words for physical spatio-temporal 
regions can do so. This is, in fact, merely a way of stating that 
physical spatio-temporal position is relative, not absolute. 

Assuming this correct so far, the question arises whether we 
need names for qualities and spatio-temporal relations. Take 
colours, for example. It may be said that they can be designated 
by wave-lengths. This leads to Carnap’s contention that there is 
nothing in physics which cannot be known to a blind man. So 
far as theoretical physics is concerned, this is obviously true. It 
is true also, up to a point, in the empirical field. We see that the 
sky is blue, but a race of blind men could devise experiments 
showing that transverse waves of certain wave-lengths proceed 
from it, and this is just what the ordinary physicist qud physicist, 
is concerned to assert. The physicist, however, does not trouble 
to assert, and the blind man cannot assert, the proposition : “When 


human knowledge: its scope and limits 

light of a certain frequency strikes a normal eye, it causes a 
sensation of blue. ,, This statement is not a tautology; it was a 
discovery, made many thousands of years after words for “blue” 
had been in common use. 

The question whether the word “blue” can be defined is not 
easy. We might say: “Blue” is the name of colour-sensations 
caused by light of such-and-such frequencies. Or we might say: 
“Blue” is the name of those shades of colour which, in the spectrum, 
come between violet and green. Either of these definitions might 
enable us to procure for ourselves a sensation of blue. But when 
we had done so, we should be in a position to say: “So that is 
blue .” This would be a discovery, only to be made by actually 
experiencing blue. And in this statement, I should say, “that” 
is in one sense a proper name, though of that peculiar sort that 
I call “egocentric”. 

We do not usually give names to smells and tastes, but we 
could do so. Before going to America, I knew the proposition “the 
smell of a skunk is disagreeable”. Now I know the two proposi- 
tions: “ that is the smell of a skunk”, and “ that is disagreeable”. 
Instead of “that”, we might use a name, say “pfui”, and should 
do so if we often wished to speak of the smell without men- 
tioning skunks. But to any one who had not had the requisite 
experience, the name would be an abbreviated description, not 
a name. 

I conclude that names are to be applied to what is experienced, 
and that what is experienced does not have, essentially and neces- 
sarily, any such spatio-temporal uniqueness as belongs to a space- 
time region in physics. A word must denote something that can 
be recognized , and space-time regions, apart from qualities, cannot 
be recognized, since they are all alike. They are in fact logical 
fictions, but I am ignoring this for the moment. 

There are occurrences that I experience, and I believe there 
are others that I do not experience. The occurrences that I 
experience are all complex, and can be analysed into qualities 
with spatial and temporal relations. The most important of these 
relations are compresence, contiguity, and succession. The words 
that we use to designate qualities are not precise; they all have 
the sort of vagueness that belongs to such words as “bald” and 
“fat”. This is true even of the words that we are most anxious 
to make precise, such as “centimetre” and “second”. Words 



designating qualities must be defined ostensively, if we are to be 
able to express observations; as soon as we substitute a verbal 
definition, we cease to express what is observed. The word “blue,” 
for instance, will mean “a colour like that ”, where that is a blue 
patch. How like that it must be to be blue, we cannot state with 
any precision. 

This is all very well, but how about such words as “this” and 
“that”, which keep on intruding themselves? We think of the 
word “this” as designating something which is unique, and can 
only occur once. If, however, “this” denotes a bundle of corn- 
present qualities, there is no logical reason why it should not 
recur. I accept this. That is to say, I hold that there is no 
class of empirically known objects such that, if x is a mem- 
ber of the class, the statement “ x precedes x ” is logically 

We are accustomed to think that the relation “precedes” is 
asymmetrical and transitive . 1 “Time” and “event” are both con- 
cepts invented with a view to securing these properties to the 
relation “precedes”. Most people have discarded “time” as some- 
thing distinct from temporal succession, but they have not dis- 
carded “event”. An “event” is supposed to occupy some con- 
tinuous portion of space- time, at the end of which it ceases, and 
cannot recur. It is clear that a quality, or a complex of qualities, 
may recur; therefore an “event”, if non-recurrence is logically 
necessary, is not a bundle of qualities. What, then, is it, and how 
is it known? It will have the traditional characteristics of sub- 
stance, in that it will be a subject of qualities, but not defined 
when all its qualities are assigned. And how do we know that 
there is any class of objects the members of which cannot recur? 
If we are to know this, it might seem that it must be a case of 
synthetic a priori knowledge, and that if we reject the synthetic 
a priori , we must reject the impossibility of recurrence. We shall, 
of course, admit that, if we take a sufficiently large bundle of 
qualities, there will be no empirical instance of recurrence. Non- 
recurrence of such bundles may be accepted as a law of physics, 
but not as something necessary. 

The view that I am suggesting is that an “event” may be 
defined as a complete bundle of compresent qualities, i.e. a bundle 

1 I.e. that if A precedes B, B does not precede A, and if A precedes B 
and B precedes C, then A precedes C. 

97 G 

human knowledge: its scope and limits 

having the two properties (a) that all the qualities in the bundle 
are compresent, (6) that nothing outside the bundle is compresent 
with every member of the bundle. I assume that, as a matter of 
empirical fact, no event recurs; that is to say, if a and b are 
events, and a is earlier than b y there is some qualitative difference 
between a and b . For preferring this theory to one which makes 
an event indefinable, there are all the reasons commonly alleged 
against substance. If two events were exactly alike, nothing could 
ever lead us to suppose that they were two. In taking a census, 
we could not count one apart from the other, for, if we did, that 
would be a difference between them. And from the standpoint 
of language, a word must denote something that can be recog- 
nized, and this requires some recognizable quality. This leads 
to the conclusion that words such as “Napoleon” can be defined, 
and are therefore theoretically unnecessary; and that the same 
thing would be true of words designating events, if we were 
tempted to invent such words. 

I conclude that, if we reduce our empirical vocabulary to a 
minimum, thereby excluding all words that have verbal definitions, 
we shall still need words for qualities, compresence, succession, 
and observed spatial relations, i.e. spatial relations which can be 
discriminated within a single sensible complex. It is an empirical 
fact that, if we form a complex of all the qualities that are all 
compresent with each other, this complex is found, so far as our 
experience goes, not to precede itself, i.e. not to recur. In forming 
the time-series, we generalize this observed fact. 

The nearest approach to proper names in such a language 
will be the words for qualities and complexes of compresent 
qualities. These words will have the syntactical characteristics of 
proper names, but not certain other characteristics that we expect, 
for example that of designating a region which is spatio-tem- 
porally continuous. Whether, in these circumstances, such words 
are to be called “names”, is a matter of taste, as to which I 
express no opinion. What are commonly called proper names — 
e.g. “Socrates” — can, if I am right, be defined in terms of qualities 
and spatio-temporal relations, and this definition is an actual 
analysis. Most subject-predicate propositions, such as “Socrates 
is snub-nosed”, assert that a certain quality, named by the 
predicate, is one of a bundle of qualities named by the subject — 
this bundle being a unity in virtue of compresence and causal 



relations. Proper names in the ordinary sense, if this is right, are 
misleading, and embody a false metaphysic. 

Note . — The above discussion of proper names is not intended 
to be conclusive. The subject will be resumed in other contexts, 
especially in Part IV, Chapter VIII. 


Chapter IV 


I give the name “egocentric particulars” to words of which 
the meaning varies with the speaker and his position in time 
and space. The four fundamental words of this sort are “I”, 
“this”, “here”, and “now”. The word “now” denotes a different 
point of time on each successive occasion when I use it; the word 
“here” denotes a different region of space after each time when 
I move; the word “I” denotes a different person according to 
who it is that utters it. Nevertheless there is obviously some sense 
in which these words have a constant meaning, which is the reason 
for the use of the words. This raises a problem, but before con- 
sidering it let us consider what other words are egocentric, and 
especially what words are really egocentric although intended 
not to be so. 

Among obviously egocentric words are “near” and “far”, 
“past”, “present”, and “future”, “was”, “is”, and “will be”, and 
generally all forms of verbs involving tense. “This” and “that” 
are obviously egocentric; in fact, “this” might be taken as the 
only egocentric word not having a nominal definition. We could 
say that “I” means “the person experiencing this”, “now” means 
“the time ol this”, and “here” means “the place of this”. The 
word “this” is, in a sense, a proper name, but it differs from true 
proper names in the fact that its meaning is continually changing. 
This does not mean that it is ambiguous, like (say) “John Jones”, 
which is at all times the proper name of many different men. 
Unlike “John Jones”, “this” is at each moment the name of only 
one object in one person’s speech. Given the speaker and the 
time, the meaning of “this” is unambiguous, but when the speaker 
and the time are unknown we cannot tell what object it denotes. 
For this reason, the word is more satisfactory in speech than in 
print. If you hear a man say “this is an age of progress”, you 
know what age he refers to ; but if you read the same statement 
in a book it may be what Adam said when he invented the spade 
or what was said by any later optimist. You can only decide what 
the statement means by finding out when it was written, and in 



this sense its meaning is not self-contained but requires elucidation 
by extraneous information. 

One of the aims of both science and common sense is to replace 
the shifting subjectivity of egocentric particulars by neutral public 
terms. “I” is replaced by my name, “here” by latitude and 
longitude, and “now” by date. Suppose I am walking with a 
friend on a dark night, and we lose touch with each other: he 
calls out “where are you?” and I reply “here I am”. Science will 
not accept such language; it will substitute “At 11.32 p.m. on 
January 30, 1946, B.R. was at longitude 4 0 3' 29" W. and at 
latitude 53 0 16' 14" N”. This information is impersonal: it gives 
a prescription by which a qualified person who possesses a sextant 
and a chronometer, and has the patience to wait for a sunny day, 
can determine where I was, which he may proclaim in the words 
“here is where he was”. If the matter is of sufficient importance, 
say in a trial for murder, this elaborate procedure may be worth 
the trouble it involves. But its appearance of complete imper- 
sonality is in part deceptive. Four items are involved: my name, 
the date, latitude, and longitude. In regard to each of these there 
is an element of egocentricity which is concealed by the fact that, 
for most purposes, it has no practical importance. 

From a practical point of view, the impersonality is complete. 
Two competent persons, given time and opportunity, will both 
accept or both reject a statement of the form: “At time t y A was 
at longitude B, latitude C.” Let us call this statement “P”. There 
is a procedure for determining date, latitude, and longitude, which, 
if correctly observed, leads different people to the same result, 
in the sense that, if both say truly “he was here five minutes 
ago”, they must be in each other’s presence. This is the essential 
merit of scientific terminology and scientific technique. But when 
we examine closely the meanings of our scientific terms we find 
that the subjectivity we sought to avoid has not been wholly 

Let us begin with my name. We substitute “B.R.” for “I” or 
“you” or “he”, as the case may be, because “B.R.” is a public 
appellation, appearing on my passport and my identity card. If 
a policeman says “who are you?” I might reply by saying “look I 
this is who I am”, but this information is not what the policeman 
wants, so I produce my identity card and he is satisfied. But 
essentially I have only substituted one sensible impression for 


human knowledge: its scope and limits 


another. In looking at the identity card the policeman acquires 
a certain visual impression, which enables him to say “the name 
of the accused is B.R.” Another policeman, looking at the same 
identity card, will utter what is called the “same” sentence, that 
is to say, he will emit a series of noises closely similar to those 
emitted by the first policeman. It is this similarity, mistakenly 
regarded as identity, which is the merit of the name. If the two 
policemen had had to describe my appearance, the first, delaying 
me at the end of an all-day walk in the rain, might say “he was 
a furious red-faced tramp”, while the other might say “he was 
a benign old gentleman in evening dress”. The name has the 
merit of being less variable, but it remains something known only 
through the sensible impressions of individuals, of which no two 
are exactly alike. We always come back to “this is his name”, 
where this is a present occurrence. Or rather, to be exact, “his 
name is a class of sensible occurrences all very similar to this”. 
We secure, by our procedure, a method of providing sets of 
closely similar occurrences, but we do not wholly escape from 

There is involved here a principle of considerable scope and 
importance, which deserves a more detailed exposition, to which 
we must now devote ourselves. 

Let us begin with a homely illustration. Suppose you are 
acquainted with a certain Mrs. A, and you know that her mother, 
whom you have never met, is called Mrs. B. What does the name 
“Mrs. B” mean for you? Not what it means for those who know 
her, still less what it means for her herself. It must mean some- 
thing definable in terms of your experience, as must every word 
that you can use understandingly. For every word that you can 
understand must either have a nominal definition in terms of 
words having ostensive definitions, or must itself have an osten- 
sive definition; and ostensive definitions, as appears from the 
process by which they are effected, are only possible in relation 
to events that have occurred to you. Now the name “Mrs. B” 
is something that you have experienced; therefore when you speak 
of Mrs. B you may be mentally defining her as “the lady whose 
name is ‘Mrs. B’ ”. Or, if one were to concede (what would not 
be strictly accurate) that you are acquainted with Mrs. A, you 
might define “Mrs. B” as “the mother of Mrs. A”. In this way, 
although Mrs. B is outside your experience, you can interpret 



sentences in which her name occurs in such a way that your lack 
of experience does not prevent you from knowing if the sentences 
are true. 

We can now generalize the process involved in the above illus- 
tration. Suppose there is some object a which you know by 
experience, and suppose you know (no matter how) that there is 
just one object to which a has a known relation R, but there is 
no such object in your experience. (In the above case, a is a 
Mrs. A, and R is the relation of daughter to mother.) You can 
then give a name to the object to which a has the relation R; 
let the name be “6”. (In our illustration it was “Mrs. B”.) It 
then becomes easy to forget that b is unknown to you although 
you may know multitudes of true sentences about b. But in fact, 
to speak correctly, you do not know sentences about b ; you know 
sentences in which the name u b" is replaced by the phrase “the 
object to which a has the relation R”. You know also that there 
are sentences about the actual object b which are verbally iden- 
tical with those that you know about the object to which a has 
the relation R — sentences pronounced by other people in which 
occurs as a name — but although you can describe these 
sentences, and know (within common-sense limits) which are true 
and which false, you do not know the sentences themselves. You 
may know that Mrs. A’s mother is rich, but you do not know 
what Mrs. B knows when she says “I am rich”. 

The result of this state of affairs is that our knowledge seems 
to extend much further beyond our experience than it actually 
does. We may perhaps distinguish, in such cases as we have been 
considering, between what we can assert and what we intend. 
If I say “Mrs. B is rich”, I intend something about Mrs. B her- 
self, but what I actually assert is that Mrs. A has a rich mother. 
Another person may know of Mrs. B, not as the mother of Mrs. A, 
but as the mother of another daughter Mrs. C. In that case, when 
he says “Mrs. B is rich”, he means “Mrs. C has a rich mother”, 
which is not what I meant. But we both intend to say something 
about Mrs. B herself, though in this neither of us is successful. 
This does not matter in practice, as the things we respectively 
say about Mrs. A’s mother or Mrs. C’s mother would be true 
of Mrs. B if only we could say them. But although it does not 
matter in practice, it matters greatly in theory of knowledge. For 
in fact everybody except myself is to me in the position of Mrs. B ; 


HUMAN knowledge: its scope and limits 

of our supposed subjective prison. In this process of escape the 
interpretation of egocentric particulars is a very essential step. 

Before attempting a precise account of egocentric words, let us 
briefly survey the picture of the world to which subsequent 
discussions will lead us. 

There is one public space, namely the space of physics, and 
this space is occupied by public physical objects. But public 
space and public objects are not sensible; they are arrived at by 
a mixture of inference and logical construction. Sensible spaces 
and sensible objects differ from one person to another, though 
they have certain affinities to each other and to their public 

There is one public time , 1 in which not only physical events, 
but mental events also, have their place. There are also private 
times, which are those given in memory and expectation. 

My whole private space is “here” in physical space, and my 
whole private time is “now” in public time. But there are also 
private “heres” and “nows” in private spaces and times. 

When your friend calls out in the dark “where are you?” and 
you answer “here I am”, the “here” is one in physical space, 
since you are concerned to give information which will help 
another to find you. But if, when alone, you are looking for a lost 
object, and on finding it you exclaim “here it is”, the “here” may 
be either in public space or in your private space. Of course 
ordinary speech does not distinguish between public and private 
space. Broadly, “here” is where my body is — my physical body if 
I mean “here” in physical space, and my percept of my body if 
I mean “here” in my private space. But “here” may be much 
more narrowly localized, for instance if you are pointing out a 
thorn in your finger. One might say (though this would not quite 
accord with usage) that “here” is the place of whatever sensible 
object is occupying my attention. This, though not quite the 
usual meaning of the word, is the concept which most needs 
discussing in connection with the word “here”. 

“Now” has a similar two-fold meaning, one subjective and one 
objective. When I review my life in memory, some of the things 

1 This is subject to limitations connected with relativity. But as lan- 
guage and theory of knowledge are concerned with inhabitants of the 
earth, these may be ignored, since no two people have a relative velocity 
comparable to that of light. 



I remember seem a long time ago, others more recent, but all 
are in the past as compared with present percepts. This “past- 
ness”, however, is subjective: what I am remembering, I re- 
member now , and my recollecting is a present fact. If my memory 
is veridical, there was a fact to which my recollection has a certain 
relation, partly causal, partly of similarity ; this fact was objectively 
in the past. I maintain that, in addition to the objective relation 
of before-and-after, by which events are ordered in a public 
time-series, there is a subjective relation of more-or-less-remote, 
which holds between memories that all exist at the same objective 
time. The private time-series generated by this relation differs 
not only from person to person, but from moment to moment in 
the life of any one person. There is also a future in the private 
time-series, which is that of expectation. Both private and public 
time have, at each moment in the life of a percipient, one peculiar 
point, which is, at that moment, called “now”. 

It is to be observed that “here” and “now” depend upon 
perception; in a purely material universe there would be no 
“here” and “now”. Perception is not impartial, but proceeds 
from a centre; our perceptual world is (so to speak) a perspective 
view of the common world. What is near in time and space 
generally gives rise to a more vivid and distinct memory or percept 
than what is far. The public world of physics has no such centre 
of illumination. 

In defining egocentric particulars, we may take “this” as 
fundamental, in a sense in which “this” is not distinguished from 
“that”. I shall attempt an ostensive definition of “this”, and 
thence a nominal definition of the other egocentric particulars. 

^This” denotes whatever, at the moment when the word is 
used, occupies the centre of attention. With words which are not 
egocentric, what is constant is something about the object 
indicated, but “this” denotes a different object on each occasion 
of its use: what is constant is not the object denoted, but its 
relation to the particular use of the word. Whenever the word is 
used, the person using it is attending to something, and the word 
indicates this something. When a word is not egocentric, there is 
no need to distinguish between different occasions when it is 
used, but we must make this distinction with egocentric? words, 
since what they indicate is something having a given relation to 
the particular use of the word. ^ 


HUMAN knowledge: its scope and limits 

We may define “I” as “the person attending to this”, “now” 
as “the time of attending to this”, and “here” as “the place of 
attending to this”. We could equally well take “here-now” as 
fundamental; then “this” would be defined as “what is here- 
now”, and “I” as “what experiences this”. 

Can two persons experience the same “this”, and if so, in what 
circumstances? I do not think this question can be decided by 
logical considerations : a priori , either answer would be possible. 
But taking the question empirically, it has an answer. When the 
“this” concerned is what common sense takes to be a percept of 
a physical object, difference of perspective makes a difference in 
the percept unavoidable, if the same physical object is concerned 
in the two cases. Two people looking at one tree, or listening to 
the song of one bird, are having somewhat different percepts. 
But two people looking at different trees might, theoretically, have 
exactly similar percepts, though this would be improbable. Two 
people may see exactly the same shade of colour, and are likely 
to do so if each is looking at a continuous band of colours, e.g. 
those of the rainbow. Two people looking at a square table will 
not see exactly similar quadrilaterals, but the quadrilaterals they 
see will have certain geometrical properties in common. 

It thus appears that two people are more likely to have the 
same “this” if it is somewhat abstract than if it is fully concrete. 
In fact, broadly speaking, every increase of abstractness diminishes 
the difference between one person’s world and another’s. When 
we come to logic and pure mathematics, there need be no difference 
whatever : two people can attach exactly the same meaning to the 
word “or” or the word “371 ,294”. This is one reason why physics, 
in its endeavour to eliminate the privacy of sense, has grown 
progressively more abstract. This is also the reason for the view, 
which has been widely held by philosophers, that all true know- 
ledge is intellectual rather than sensible, and that the intellect 
iberates while the senses keep us in a personal prison. In such 
views there is an element of truth, but no more, except where 
logic and pure mathematics are concerned; for in all empirical 
knowledge liberation from sense can be only partial. It can, 
however, be carried to the point where two men’s interpretations 
of a given sentence are nearly certain to be both true or both false. 
The securing of this result is one of the aims (more or less un- 
conscious) governing the development of scientific concepts. 


Chapter V 


W E have been concerned hitherto with what may be 
called the “exclamatory” use of language, when it is 
used to denote some interesting feature of a man's 
present experience. So long as this use alone is in question, a 
single word can function as a sentence in the indicative. When 
Xenophon's Ten Thousand exclaimed “Sea! Sea!” they were 
using the word in this way. But a single word may also be used 
in other ways. A man found dying of thirst in the desert may 
murmur “water!” and is then uttering a request or expressing a 
desire; he may see a mirage and say “water?”; or he may see a 
spring and assert “water”. Sentences are needed to distinguish 
between these various uses of words. They are needed also — and 
this is perhaps their main use — to express what may be called 
“suspended reactions”. Suppose you intend to take a railway 
journey to-morrow, and you look up your train to-day : you do not 
propose, at the moment, to take any further action on the know- 
ledge you have acquired, but when the time comes you will 
behave in the appropriate manner. Knowledge, in the sense in 
which it does not merely register present sensible impressions, 
consists essentially of preparations for such delayed reactions. 
Such preparations may in all cases be called “beliefs”, but they 
are only to be called “knowledge” when they prompt successful 
reactions, or at any rate show themselves related to the facts with 
which they are concerned in some way which distinguishes them 
from preparations that would be called “errors”. 

It is important not to exaggerate the role of language. In my 
view, there is in pre-linguistic experience something that may be 
called “belief”, and that may be true or false; there are also, I 
should say, what may be called “ideas”. Language immensely 
increases the number and complexity of possible beliefs and ideas, 
but is not, I am convinced, necessary for the simplest beliefs and 
ideas. A cat will watch for a long time at a mousehole, with her 
tail swishing in savage expectation; in such a case, one should 
say (so I hold) that the smell of mouse stimulates the “idea” of 



the rest of what makes up an actual mouse. The objection to such 
language comes, it seems to me, from an unduly intellectualist 
conception of what is meant by the word “idea”. I should define 
an “idea” as a state of an organism appropriate (in some sense) 
to something not sensibly present. All desire involves ideas in 
this sense, and desire is certainly pre-linguistic. Belief also, in an 
important sense, exists in the cat watching the mousehole, belief 
which is “true” if there is a mouse down the hole and “false” if 

The word “mouse”, by itself, will not express the different 
attitudes of the cat while waiting for her prey and when seizing 
it; to express these different attitudes further developments of 
language are necessary. Command, desire, and narrative all 
involve the use of words describing something not sensibly 
present, and to distinguish them from each other and from the 
indicative various linguistic devices are necessary. 

Perhaps the necessity to assume “ideas” as existing ante- 
cedently to language may be made more evident by considering 
what it is that words express. The dying man in the desert who 
murmurs “water 1” is clearly expressing a state in which a dying 
animal might be. How this state should be analysed is a difficult 
question, but we all, in a sense, know the meaning of the word 
“thirst”, and we all know that what this word means does not 
depend for its existence upon there being a word to denote it. 
The word “thirst” denotes a desire for something to drink, and 
such a desire involves, in the sense already explained, the presence 
of the “idea” of drink. What would commonly be called a man's 
“mental” life is entirely made up of ideas and attitudes towards 
them. Imagination, memory, desire, thought, and belief all involve 
ideas, and ideas are connected with suspended reactions. Ideas, 
in fact, are parts of causes of actions, which become complete 
causes when a suitable stimulus is applied. They are like explosives 
waiting to be exploded. In fact, the similarity may be very close. 
Trained soldiers, hearing the word “fire!” (which already existed 
in them as an idea) proceed to cause explosions. The similarity of 
language to explosives lies in the fact that a very small additional 
stimulus can produce a tremendous effect. Consider the effects 
which flowed from Hitler’s pronouncing the word “war!” 

It is to be observed that words, when learnt, can become 
substitutes for ideas. There is a condition called “thinking of” 



this or that, say water when you are in the desert. A dog appears, 
from its behaviour, to be capable of being in this condition ; so 
does an infant that cannot yet speak. When this condition exists, 
it prompts behaviour having reference to water. When the word 
“water” is known, the condition may consist (mainly, not wholly) 
in the presence of this word, either overtly pronounced or merely 
imagined. The word, when understood, has the same causal 
efficacy as the idea. Familiar knowledge is apt to be purely verbal; 
few schoolboys go beyond the words in reciting “William the 
Conqueror 1066”. Words and ideas are, in fact, interchangeable; 
both have meaning, and both have the same kind of causal 
relations to what they mean. The difference is that, in the case of 
words, the relation to what is meant is in the nature of a social 
convention, and is learnt by hearing speech, whereas in the case 
of ideas the relation is “natural”, i.e. it does not depend upon the 
behaviour of other people, but upon intrinsic similarity and (one 
must suppose) upon physiological processes existing in all human 
beings, and to a lesser extent in the higher animals. 

“Knowledge”, which is, in most forms, connected with sus- 
pended reactions, is not a precise conception. Many of the 
difficulties of philosophers have arisen from regarding it as 
precise. Let us consider various ways of “knowing” the same fact. 
Suppose that, at 4 p.m. yesterday, I heard the noise of an explosion. 
When I heard it, I “knew” the noise in a certain sense, though 
not in the sense in which the word is usually employed. This 
sense, in spite of being unusual, cannot be discarded, since it is 
essential in explaining what is meant by “empirical verification”. 
Immediately afterwards, I may say “that was loud!” or “what 
was that noise?” This is “immediate memory”, which differs 
only in degree from sensation, since the physiological disturbance 
caused by the noise has not yet wholly subsided. Immediately 
before the explosion, if I have seen the train fired which leads to 
a charge of explosive, I may be in a state of tense expectation; 
this is, in a sense, akin to immediate memory, but directed to the 
near future. Next comes true memory : I now remember the bang 
I heard yesterday. My state is now made up of ideas (or images) 
or words, together with belief and a context which dates the 
occurrence remembered. I can imagine a bang just like , the one 
that I remember, but when I do this, belief and dating are absent. 
(The word “belief” is one which I shall discuss later.) Imagined 


HUMAN knowledge: its scope and limits 

events are not included in knowledge or error, because of the 
absence of belief. 

Sensation, immediate expectation, immediate memory, and true 
memory all give knowledge which is, in some degree and with 
appropriate limitations, independent of extraneous evidence. But 
most of the knowledge of people with any degree of education is 
not of any of these kinds. We know what we have been told or 
have read in books or newspapers; here words come first, and it 
is often unnecessary to realize what the words mean. When I 
believe “William the Conqueror 1066”, what I am really believing 
(as a rule) is: “the words ‘William the Conqueror 1066* are true”. 
This has the advantage that the words can be made sensible 
whenever I choose; the Conqueror is dead, but his name comes 
to life whenever I pronounce it. It has also the advantage that the 
name is public and the same for all, whereas the image (if any) 
employed in thinking of William will differ from person to person, 
and is sure to be too concrete. If (e.g.) we think of him on horse- 
back, that will not suit “William was born at Falaise”, because 
he was not born on horseback. 

Sentences heard in narrative are, of course, not necessarily 
understood in this purely verbal manner; indeed a purely verbal 
understanding is essentially incomplete. A child reading an 
exciting adventure story will “live through 5 ’ the adventures of the 
hero, particularly if the hero is of about the same age as the reader. 
If the hero leaps a chasm, the child’s muscles will grow taut; if 
the hero sees a lion about to spring, the child will hold his breath. 
Whatever happens to the hero, the child’s physiological condition 
is a reproduction, on a smaller scale, of the physiological condition 
of the hero. In adult life, the same result can be produced by good 
writing. When Shakespeare’s Antony says “I am dying, Egypt, 
dying”, we experience something which we do not experience 
when we see in The Times a notice of the death of some person 
unknown to us. One difference between poetry and bald statement 
is that poetry seeks to take the reader behind the words to what 
they signify. 

The process called “verification” does not absolutely necessitate 
(but often involves) an imaginative understanding of words, but 
only a comparison of words used in advance with words used 
when the fact concerned becomes sensible. You say “this litmus 
paper will turn red”; I, later, say “this litmus paper has turned 



red”. Thus I need only pass outside the purely verbal region when 
I use a sentence to express a present sensible fact. 

“Knowledge” is a vague concept for two reasons. First, because 
the meaning of a word is always more or less vague except in logic 
and pure mathematics ; and second, because all that we count as 
knowledge is in a greater or less degree uncertain, and there is 
no way of deciding how much uncertainty makes a belief un- 
worthy to be called “knowledge”, any more than how much loss 
of hair makes a man bald. 

“Knowledge” is sometimes defined as “true belief”, but this 
definition is too wide. If you look at a clock which you believe 
to be going, but which in fact has stopped, and you happen to 
look at it at a moment when it is right, you will acquire a true 
belief as to the time of day, but you cannot be correctly said to 
have knowledge. The correct definition of “knowledge” need not 
concern us at the moment ; what concerns us now is belief. 

Let us take some simple sentence expressing something that is 
or may be a sensible fact, such as “a loud bang is (or has been, 
or will be) taking place”. We will suppose it a fact that such a 
bang occurs at a place P at time t y and that the belief to be con- 
sidered refers to this particular bang. That is to say we will amend 
our sentence to “a loud bang occurs at place P at time t y \ We 
will call this sentence S. What sort of thing is happening to me 
when I believe this sentence, or rather when I believe what it 
expresses ? 

There are a number of possibilities. First, I may be at or near 
the place P at the time t , and may hear the bang. In that case, at 
time t I have direct sensible knowledge of it; ordinary language 
would hardly call this “belief”, but for our purposes it is better 
to include it in the scope of the word. Obviously this sort of 
knowledge does not require words. No more does the immediate 
memory that subsists while I am still shaken by the noise. But 
how about more remote memory? Here, also, we may have no 
words, but an auditory image accompanied by a feeling which 
could be (but need not be) expressed in the words “that occurred”. 
Immediate expectation also does not need words. When you 
watch a door about to be slammed by the wind, your body and 
mind are in a state of expectation of noise, and if no noise Resulted 
you would experience a shock of surprise. This immediate 
expectation is different from our ordinary expectations about 



HUMAN knowledge: its scope and limits 

events that are not imminent. I expect that I shall get up to- 
morrow morning, but my body is not in that unpleasant condition 
in which it will be to-morrow morning when I am expecting to 
get up in a moment. I doubt whether it is possible, without words, 
to expect any event not in the immediate future. This is one of 
the differences between expectation and memory. 

Belief about something outside my own experience seems 
usually only possible through the help of language, or some 
rudimentary beginning of language. Sea-gulls and cannibals have 
a “food-cry”, which in the cannibals is meant to give information, 
but in the sea-gulls may be a spontaneous expression of emotion, 
like a groan when the dentist hurts you. A noise of this sort is a 
word to the hearer, but not to the utterer. An animars behaviour 
may be affected by signs which have no analogy with language, 
for instance when it is in search of water in an unknown region. 
If a thirsty animal runs persistently down into a valley, I should 
be inclined to say that it “believes” there is water there, and in 
such a case there would be non-verbal belief in something that is 
as yet outside the animal’s experience. However, I do not wish 
to become involved in a controversy as to the meaning of words, 
so I will not insist upon the view that such behaviour shows 

Among human beings, the usual way of acquiring beliefs as to 
what has not been, and is not just about to be, experienced is 
through verbal testimony. To revert to our sentence S, some 
person whom we believe to be truthful pronounces it in our 
presence, and we then believe what the sentence asserts. I want 
to inquire what is actually occurring in us while we are believing 
the sentence. 

We must, of course, distinguish a belief as a habit from the 
same belief when it is active. This distinction is necessary in 
regard to all habits. An acquired habit consists in the fact that a 
certain stimulus, whenever it occurs, now produces a certain 
reaction which it did not produce in the animal in question until 
the animal had had certain experiences. We must suppose that, 
even in the absence of the stimulus concerned, there is some 
difference between an animal that has a certain habit and one 
that lacks it. A man who understands the word “fire” must differ 
in some way from a man who does not, even when he is not hearing 
the word. We suppose the difference to be in the brain, but its 



nature is hypothetical. However, it is not a habit as a permanent 
character of an organism that concerns us, but the active habit, 
which is only displayed when the appropriate stimulus is applied. 
In the case we are investigating, the stimulus is the sentence S; 
or rather, since the sentence may have been never heard before, 
and may therefore have had no chance to generate a habit, the 
stimulus is the succession of the words composing S, each of 
which, we are supposing, is familiar to the hearer, and has already 
generated the habit which constitutes understanding of its 

It may happen, when we hear a sentence, that we do not trouble 
to think what it means, but merely believe “this sentence is true”. 
With certain kinds of sentences, this is the usual reaction; for 
example, when we are told someone’s address and we only wish 
to write to him. If we wish to go and see him, the meaning of the 
words becomes important, but for sending him a letter the words 
alone are sufficient. When we believe “this sentence is true”, we 
are not believing what the sentence asserts; if the sentence is in 
a language unknown to us, we may believe that it is true without 
being able to find out what it asserts — for example, if it is a 
sentence in a Greek Testament and we know no Greek. I shall 
therefore ignore this case, and consider what happens when, 
hearing S, we believe what S asserts. 

Let us somewhat simplify the sentence, and suppose that, when 
I am walking with a friend, he says: “There was an explosion 
here yesterday.” I may believe him, or understand him without 
believing. Let us suppose that I believe him, and that I believe 
what his words assert, not merely that the words are true. The 
most important word in the sentence is “explosion”. This word, 
when I am actively understanding it, rouses in me faint imitations 
of the effects of hearing an actual explosion — auditory images, 
images of nervous shock, etc. Owing to the word “here”, these 
images are combined in my mental picture with the surrounding 
scenery. Owing to the word “yesterday”, they are combined with 
recollections of yesterday’s experiences. All this, so far, is involved 
in understanding the sentence, whether or not it is believed. I 
incline to the view that believing a sentence is a simpler occurrence 
than understanding without belief ; I think the primitive .reaction 
is belief, and that understanding without belief involves inhibition 
of the impulse to belief. What distinguishes belief is readiness foi 

n 5 

human knowledge: its scope and limits 

any action that may be called for if what is asserted is a fact. 
Suppose for instance that an acquaintance of mine has dis- 
appeared, and is known to have been hereabouts yesterday, then 
belief may prompt me to search for signs of his remains, which I 
shall not do if I understand without believing. If no such action 
is called for, there is at least the action of repeating what I have 
been told whenever it may seem appropriate to do so. 

From all this it appears that, when I believe what a certain 
sentence asserts, the words, having had their intended effect, 
need no longer be present to me. All that need exist is a state of 
mind and body appropriate to the fact that the sentence asserts. 

It is an error to suppose that beliefs consist solely in tendencies 
to actions of certain kinds. Let us take an analogy; a belief may 
be compared to a cistern plus a pipe plus a tap. The tap can be 
turned on, and the belief can influence action, but neither happens 
without an additional stimulus. When a man is believing some- 
thing, there must exist in him cither appropriate words or 
appropriate images, or, at the very least, appropriate muscular 
adjustments. Any of these, given certain additional circumstances 
(which correspond to turning on the tap), will produce action, 
and this action may be such as to show an outside observer what 
it is that is being believed; this is particularly the case if the 
action consists in pronouncing appropriate words. The impulse 
to action, given the right stimulus, is inherent in the presence of 
words, images, or muscular adjustments. To entertain an idea 
vividly and not act upon it is difficult. If, alone at night, you read 
a story in which a man is stabbed in the back, you will have an 
impulse to press your chair tight against the wall. Booth the actor 
(the brother of Lincoln’s assassin), on one occasion when he was 
playing Macbeth under the influence of liquor, refused to be 
killed, and chased Macduff murderously all through the stalls. 
It is unwise to read a ghost story just before walking through a 
churchyard at midnight. As these examples show, when an idea 
is entertained without belief, the impulse to belief is not absent, 
but is inhibited. Belief is not something added to an idea previ- 
ously merely entertained, but something subtracted from an 
idea, by an effort, when the idea is considered without being 

Another example is the difficulty that uneducated people feel 
about hypotheses. If you say “let us suppose so-and-so and see 



what comes of the supposition”, they will tend either to believe 
what you suppose, or to think that you are wasting your time. 
For this reason, reductio ad absurdum is a form of argument that 
is repugnant to those who are not familiar with logic or mathe- 
matics ; if the hypothesis is going to be proved false, they cannot 
make themselves hypothetically entertain it. 

I do not wish to exaggerate the scope of pre-linguistic belief: 
only very simple and primitive matters can be dealt with in the 
absence of words. Words are public, permanent (when written), 
and capable of being created at will. These merits make it possible 
to have more complicated habits based on words than any that 
could be based on wordless ideas or images. By acquiring verbal 
habits we can prepare ourselves for actual situations when they 
arise. What is more, knowledge can be externalized in books of 
reference, and need then only exist in human beings when it is 
wanted. Consider the telephone book: no one wants to know all 
its contents, or indeed any except at certain moments. The people 
who compile the book may never use it, and the immense majority 
of those who use it have had no hand in compiling it. This kind 
of socialized potential knowledge is only rendered possible by 
language, in fact by written language. All that the user of the 
telephone needs to know is a simple prescription for deriving 
appropriate action from the appropriate entry in the book. By 
such devices we diminish enormously the amount of knowledge 
that it is necessary to carry in our heads. 

All generalized knowledge is of this sort. Suppose the geography 
book tells me that Semipalatinsk is a province and city of Central 
Asia, in the territory of the U.S.S.R. This knowledge will remain 
purely verbal unless I have occasion to go to Semipalatinsk, but 
if this should happen there are rules by which the words of the 
book show me how to produce desired experiences. In such a 
case I may be said to understand the words if I know what action 
they prescribe when I have desires connected with what the 
words mean, or, in an extreme case, merely a desire to know what 
the words mean. You may feel a longing to see the Altai Mountains, 
knowing nothing about them except that that is their name. In 
that case, the guide book shows you what you must do in order 
to know the proposition: “ These are the Altai Mountains. v When 
you have learnt arithmetic you can deal with all the innumerable 
occasions on which you have to count your change in shops, but 

n 7 

HUMAN knowledge: its scope and limits 

in learning arithmetic you need not be thinking of its applications. 
In 9uch ways the province of purely verbal knowledge becomes 
wider and wider, and at last it becomes easy to forget that verbal 
knowledge must have some relation to sensible experience. But 
except through such relation we cannot define empirical truth and 
falsehood, and to forget it is therefore fatal to any hope of a sound 


Chapter VI 

I want in this chapter to consider sentences as opposed to 
words, and to ask in what consists the understanding of words 
that do not denote objects, but occur only as parts of sentences. 
We saw that the one word “water” may be used to express what, 
if fully expressed, requires different sentences. It may mean 
“here is water”; it may mean “I want water”; it may, if pro- 
nounced with an interrogative inflexion, mean “is this water?” 
Obviously such ambiguity is not desirable, especially in writing, 
where differences of inflexion are difficult to indicate. We there- 
fore need such words as “here is”, “I want”, “is this”. It is the 
function of such words that forms the theme of this chapter. 

Consider the following sentences: “there is fire here”, “there 
was fire here”, “there will be fire here”, “is there fire here?” 
“I want fire here”, “there is no fire here”. These sentences are 
respectively present, past, future, interrogative, optative, and 
negative, but all deal with the same object, namely fire. 

The word “fire” may be caused in me in various ways. When it 
is caused by the sensible presence of fire, I communicate the fact 
by the sentence “there is fire here”; when by the memory, by 
the sentence “there was fire here”. But I may use this sentence, 
not to express a memory, but to report what I have been told, 
or to state an inference from charred embers. In the former case, 
the word “fire” is caused in me by my hearing the word; in the 
latter case, by my seeing something which I know to be an effect 
of fire. Thus when I say “there was fire here”, my state of mind 
may be one of several very different possibilities. In spite of these 
subjective differences, however, what I am asserting is the same 
in all the different cases. If my assertion is true, a certain occurrence 
took place here, and the occurrence in virtue of which it is true 
is the same whether the occurrence is remembered or known 
through testimony or inferred from present traces of past com- 
bustion. It is for this reason that we use the same words in these 
various cases, for a sentence in the indicative is concerned, not to 
express a state of mind (though it always does so), but to assert 
a fact other than that expressed by the sentence. But we will 


HUMAN knowledge: its scope and limits 

postpone the explicit consideration of truth and falsehood to a 
later chapter. 

Similar subjective ambiguities exist in connection with the 
sentence “there will be fire here”. In a situation in which you 
experience immediate expectation of fire, your subjective state is 
analogous to memory, except in the vital point of being directed 
to the future. But as a rule statements about the future are 
inferences. You may see a damp haystack fermenting and infer 
that it will burn, or you may have been told that at some future 
date there is to be a bonfire here. But again these various possi- 
bilities make no difference to what is asserted when you say “there 
will be fire here”. 

“Is there fire here ?” may be a form of imperative, or a suggestion 
for investigation. This sentence does not make an assertion, but 
shows a desire to be able to make one. The difference from “there 
is a fire here” is not in anything having an external reference, but 
in our attitude to what has such a reference. We may say that 
there is an “idea” called “fire-hcre-now” ; when we preface these 
words by “there is” we assert this idea, whereas when we preface 
them by “is there” we “actively consider” them, i.e. we are 
concerned to find out whether or not to assert them. I speak of 
“outward reference” in a preliminary way, as the concept is a 
difficult one, demanding considerable discussion. 

“I want fire here” is a sentence in the indicative, asserting that 
I feel a certain desire, but it is commonly used as if it were an 
expression of desire, not an assertion of it. Strictly speaking, desire 
ought to be expressed by “would there were a fire here” or “Oh 
for a fire here!” This is more easily and naturally expressed in a 
language which, like Greek, has an optative mood. The sentence 
“Oh for a fire!” asserts nothing, and is therefore neither true nor 
false. It expresses a desire, and a person hearing me pronounce it 
may infer that I feel a desire, but it does not assert that I feel a 
desire. Similarly when I say “there is a fire here” I express a 
belief, and the hearer can infer that I have this belief, but I do 
not assert that I have a belief. 

When I say “there is not a fire here”, what may be called the 
“content” is the same as w'hen I say “there is fire here”, but this 
content is denied instead of being asserted. 

Reviewing the above sentences, but omitting those referring to 
past or future, we find that, considering what they express, they 



all have the same core, namely “fire-here-now”. The ideas 
expressed by “fire”, “here”, and “now” may be called “indica- 
tive” ideas, that is to say, they can all indicate features of a 
sensible experience. Taken all together, they constitute one 
complex indicative idea. An indicative idea sometimes indicates, 
and sometimes does not; if there is a fire here now, “fire-here- 
now” indicates that fire, but if there is not a fire, “fire-here-now” 
indicates nothing. Towards an indicative idea we may have 
various attitudes: assertive, interrogative, optative, or negative. 
These attitudes are expressed by the words “there is”, “is there”, 
“oh for”, and “there is not” respectively. (I do not pretend that 
this list of possible attitudes is exhaustive.) These attitudes, which 
are expressed by the above words, can also be asserted , but we then 
need indicative words for them; the words are “belief”, “doubt”, 
“desire”, “disbelief”. This leads to new sentences, all of which 
are assertions, but about my state of mind, not about fire. The 
sentences are: “I believe there is a fire here now”, “I wonder if 
there is a fire here now”, “I hope there is a fire here now”, and 
“I disbelieve that there is a fire here now”. 

It is evident that “thcre-is”, “is-thcrc”, “oh-for”, and “there- 
is-not” should each be regarded as one word, and as expressing 
different attitudes on the part of the speaker to one and the same 
idea. It is not their function to indicate objects, as names do; the 
fact that the word “not” can be used significantly does not imply 
that there is an object called “not” in some Platonic heaven. For 
the understanding of language it is essential to realize that, while 
some necessary words mean objects, others do not. 

Words that mean objects may be called “indicative” words. I 
include among such words not only names, but words denoting 
qualities such as “white”, “hard”, “warm”, and words denoting 
perceptible relations such as “before”, “above”, “in”. If the sole 
purpose of language were to describe sensible facts, we could 
content ourselves with indicative words. But, as we have seen, 
such words do not suffice to express doubt, desire, or disbelief. 
They also do not suffice to express logical connections, e.g. “if 
that is so, I’ll eat my hat”, or “if Wilson had been more tactful, 
America would have joined the League of Nations”. Nor do they 
suffice for sentences needing such words as “all” and *' ‘some”, 
“th/e” and “a”. The significance of words of this kind can only be 
explained by explaining the significance of sentences in which 


HUMAN knowledge: its scope and limits 

they occur. When you want to explain the word “lion,” you can 
take your child to the Zoo and say “look, that’s a lion.” But there 
is no Zoo where you can show him if or the or nevertheless , for 
these are not indicative words. They are needed in sentences, but 
only in sentences not concerned exclusively with the assertion of 
single facts. It is because we need such sentences that words 
which are not indicative are indispensable. 


Chapter VII 


T he kind of external reference with which we shall be 
concerned in this chapter is not that by which experiences 
are interpreted as percepts of external objects, as when, 
for instance, a visual sensation produces in me a condition called 
“seeing a table”. This kind of external reference will be considered 
in connection with the interpretation of physics and the evidence 
for its truth. What we are now concerned with is a reference of 
one part of my mental life to another, and only derivatively to 
things not forming part of my experience. 

We are in the habit of saying that we think of so-and-so and 
that we believe in such-and-such. It is the meaning of “of” and 
“in” in such phrases that I wish to discuss, as a necessary pre- 
liminary to the definition of “truth” and “falsehood”. 

We considered in an earlier chapter the process of ostensive 
definition as the source of the meanings of words. But we then 
found that a given word can “mean” an idea as well as a sensible 
experience; this happens, notably, when the word is used to 
express a memory. When the same word can be used to denote 
an idea or a sensible experience, that is a sign that the idea is an 
idea “of” the sensible experience. But obviously the relation 
expressed by this word “of” is one which can exist independently 
of language, and is, in fact, presupposed in the use of the same 
word for an idea and a sensible experience. 

The relation with which we are concerned is perhaps seen most 
clearly in the case of memory. Suppose you have lately seen 
something horrible — say a friend run over and killed by a lorry. 
You will have a constantly recurring picture of the event in your 
mind, not only as pure imagination, but as something that you 
know actually occurred. As the dreadful swift sequence over- 
whelms you once more, you may say to yourself “yes, that really 
happened”. But in what sense can this be true? For your recol- 
lections are now , and consist of images, not of sensations, still less 
of actual motor lorries. The sense in which it is nevertheless true 
is w r hat we have to elucidate. 


HUMAN knowledge: its scope and limits 

Images occur in two ways, as imagination and as recollection. 
I have sometimes, under the influence of fatigue or fever, seen 
the faces of people of whom I was fond, not with the benign 
expression to which I was accustomed, but horribly grimacing 
and grotesque. These painful images did not command belief 
unless my temperature was high enough to cause delirium. Even 
in deliberate recollection there are often imaginative accretions 
which are not believed, but these do not count as memories. 
Whatever counts as a memory consists of images or words which 
are felt as referring to some earlier experience. Since it is clear 
that words can only express memories because a given word can 
apply both to an image (or idea) and to a sensible occurrence, it 
is clear that we must first consider non-verbal memory, with a 
view to discovering what is the relation of an idea to a sensible 
experience which leads us to use the same word for both. I shall 
therefore, for the present, exclude memories expressed in words, 
and consider only those that come as images accompanied by the 
belief or feeling that they refer to a previous occurrence. 

Suppose I am asked to describe the furniture of my room. I 
may go to my room and record what I see, or I may call up a 
picture of my room and record what I see with my mind's eye. 
If I am a good visualizer and my room is one which I have 
inhabited for some time, the two methods will give results that, 
at least in broad outline, will be indistinguishable. It is easy, in 
this way, to test the accuracy of my memory. But before it is 
tested I implicitly believe it. Some memories are not capable of 
being tested at all thoroughly, for instance if you have been the 
sole spectator of a murder; nevertheless your evidence will be 
accepted unless there is reason to suspect you of perjury. At 
present, it is not the trustworthiness of memory that concerns 
us, but the analysis of the occurrence. 

What is involved in saying that A is an “image" or “idea" of 
B ? First, there must be resemblance ; more particularly, if both 
are complex, there must be resemblance of structure. Second, B 
must play a certain definite part in the causation of A. Third, 
A and B must have certain effects in common, for example, they 
can cause the same words to occur to a person who experiences 
them. When these three relations exist, I shall say that B is the 
“prototype" of A. 

But if A is a recollection of B, something more is involved. 



For in this case A is felt or believed to be pointing to something 
other than itself, and this something is, in fact, B. We should like 
to say that A is felt to be pointing to B, but this we have no right 
to say, since B is not itself present to the person recollecting; 
what is present is only A, as B’s representative. We must say, 
therefore, that, in memory as opposed to pure imagination, there 
is the belief: “A is related to something as idea to prototype”, 
where the relation of idea to prototype is defined by the three 
characteristics mentioned in the preceding paragraph. I do not 
mean, of course, that an ordinary memory belief has the explicit- 
ness suggested by the above analysis. I mean only that, in memory, 
an idea is vaguely felt to point beyond itself, and that the above 
is an account of what may be the actual state of affairs when this 
vague feeling is justifiable. 

When B is the prototype of A, we say that A is an image “of” 
B. This is a definition of this use of the word “of”. 

It is obvious that A may be an image of B without the person 
concerned being aware of the fact. It is also obvious that A may 
have many prototypes. If I tell you I met a negro in an English 
country lane, the word “negro” may call up in your mind an 
image vaguely compounded of many negroes whom you have 
seen; in this case they must all count as prototypes of your image. 
In general, even when an image has only one prototype, it will 
usually be vaguer than its prototype. If, for example, you have 
an image of a shade of colour, various shades that you can dis- 
tinguish when sensibly present might all serve as its prototypes. 
This, incidentally, supplies an answer to Hume’s query: Could 
you imagine a shade of colour you had never seen, if it was inter- 
mediate between two very similar shades that you had seen? 
The answer is that you could not form so precise an image, even 
of a colour that you had seen, but that you could form a vague 
image, equally appropriate to the shade that you had not seen 
and to the two similar shades that you had seen. 

It will be seen that, according to the above theory, the external 
reference of an idea or image consists in a belief, which, when 
made explicit, may be expressed in the words: “this has a proto- 
type”. In the absence of such a belief (which, when it exists, is 
usually a somewhat vague feeling), although there may be 'in fact 
a prototype there is no external reference. This is the case of pure 


HUMAN knowledge: its scope and limits 

In the case of a memory- belief, if what is said to be remembered 
is an experience of the person remembering, the above kind of 
external reference is the only kind required. But as a rule there 
is also another kind, namely that which, at the beginning of this 
chapter, we declined to consider. Suppose I remember “I saw 
an elephant yesterday”. There is involved not only my experience 
of yesterday, but belief in an animal having an independent 
existence, not only when I saw it, but also before and after. All 
this depends upon animal inference in the sense to be considered 
in Part III, involving a reference external, not only to my present 
experience, but to the whole of my experience. This kind of 
external reference, however, takes us beyond the subject of the 
present chapter. 


Chapter VIII 


T ruth and falsehood, in so far as they are public, are 
attributes of sentences, either in the indicative or in the 
subjunctive or conditional. In the present chapter, which 
will consider only the simpler examples of truth, I shall confine 
myself to sentences in the indicative. In addition to sentences 
there are some other ways of making public statements, maps, for 
instance, and graphs. There are also conventional devices for 
reducing a sentence to one essential word, as is done in telephone 
books and railway time tables. But for our purposes we may, 
without any important loss of generality, confine ourselves to 
fully expressed sentences. And until we have considered logical 
words, which will be the subject of the next two chapters, we 
must confine ourselves to sentences in the indicative. 

But in order to define “truth” and “falsehood” we must go 
behind sentences to what they “express” and what they “indicate”. 

A sentence has, to begin with, a properly which I shall call 
“signification”. This is the property which is preserved in an 
accurate translation. “Two and two make four” has the same 
signification as “deux et deux font quatre”. Signification is also 
preserved when the wording is changed; c.g. “A is the husband 
of B”, “B is the wife of A”, “A is a male who is married to B”, 
“A is married to B, who is a female”, all have the same significa- 
tion. It is obvious that when two sentences have the same significa- 
tion both are true or both are false; therefore whatever dis- 
tinguishes truth from falsehood is to be sought rather in the 
signification of sentences than in sentences themselves. 

Some sentences which, at first sight, appear to be correctly 
constructed, are in fact nonsense, in the sense that they have no 
signification. Such, if interpreted literally, are “necessity is the 
mother of invention” and “procrastination is the thief of time”. 
A very important part of logical syntax consists of rules for 
avoiding nonsense in constructing sentences. But for the present 
we are concerned with sentences that are too simple to run the 
risk of being nonsensical. 

To arrive at what a sentence “signifies”, the easiest way is to 


HUMAN knowledge: its scope and limits 

ask ourselves what is in common between a sentence in one 
language and its translation into another. Suppose that on a 
given occasion I say to an Englishman “I am hot” and to a 
Frenchman “j’ai chaud”, the two sentences express the same 
state of mind and body, and are made true (or false) by the same 
fact. The signification of a sentence would thus seem to have 
two aspects: on the one hand it “expresses” the condition of the 
person uttering it, and on the other hand it points outside this 
present condition to something in virtue of which it is true or 
false. What an asserted sentence expresses is a belief ; what makes 
it true or false is a fact , which is in general distinct from the belief. 
Truth and falsehood are external relations, that is to say, no 
analysis of a sentence or a belief will show whether it is true or 
false. (This does not apply to logic and mathematics, where truth 
or falsehood, as the case may be, follows from the form of the 
sentence. But I am for the present ignoring logical truth.) Con- 
sider e.g. the sentence “I am an uncle”, and suppose you know 
that your sister in India is due to have a child, but you do not 
know whether the child has yet been born. No analysis of the 
sentence or of your state of mind will show whether the sentence 
is true or false, since its truth or falsehood depends upon events 
in India as to which you are in ignorance. But although under- 
standing the sentence does not enable you to know whether it is 
true or false, it does enable you to know what sort of fact would 
make it true and what sort would make it false; this, therefore, is 
part of the signification of a sentence, or is at least inseparably 
connected with the signification, although the actual truth or 
falsehood (as the case may be) is not. 

If “truth” and “falsehood” had been defined, we could say 
that two sentences are to have, by definition, the same “significa- 
tion” if whatever possible state of affairs makes one of them true 
also makes the other true, and vice versa. But, as we shall see, it 
is not clear that “truth” and “falsehood” can be defined without 
first defining “signification”. 

There are, we said, two sides to signification, which we may 
call subjective and objective respectively. The subjective side has 
to do with the state of the person uttering the sentence, while the 
objective side has to do with what would make the sentence true 
or false. Let us begin by considering the subjective side. 

When we say that a sentence is true, we mean that a person 


truth: elementary forms 

asserting it will be speaking truly. A person may pronounce a 
sentence without intending to assert it: when an actor says “this 
is I, Hamlet the Dane”, no one believes him, but no one accuses 
him of lying. This shows that the subjective side in the analysis 
of signification is essential. When we say that a sentence is “true”, 
we mean to say something about the state of mind of a person 
uttering or hearing it with belief. It is in fact primarily beliefs 
that are true or false ; sentences only become so through the fact 
that they can express beliefs. It is therefore in beliefs that the 
subjective side of the signification of sentences is to be sought. 

We may say that two sentences have the same signification if 
they express the same belief. But we must, having said this, 
explain in what sense two people (or one person at different 
times) can have the same belief, and by what tests we can dis- 
cover when this is the case. For practical purposes we may say 
that two people who speak the same language have the same 
belief if they accept the same sentence as expressing it ; and when 
two people speak different languages, their beliefs are the same 
if a competent interpreter regards the sentence in which one of 
them expresses his belief as a translation of that used by the other. 
But this criterion is not theoretically sufficient, since infants that 
cannot speak must be allowed to have beliefs, and so (I should 
say) must animals. 

“Belief”, as I wish to use the word, denotes a state of mind 
or body or both, in which an animal acts with reference to some- 
thing not sensibly present. When I go to the station in expectation 
of finding a train, my action expresses a belief. So does the action 
of a dog excited by the smell of fox. So does that of a bird in a 
room, which flies against the window panes in the hope of getting 
out. Among human beings, the only action by which a belief 
is expressed is, very often, the pronouncing of appropriate words. 

It will be seen that, according to the above definition of “belief”, 
it is closely connected with meaning and with ostensive definition. 
Words have “meaning” when there is an association or a con- 
ditioned reflex connecting them with something other than them- 
selves — this, at least, applies to indicative words. I say “look, 
there’s a fox”, and you act as you would do if you smelt a fox. 
I say “fox” when I see a fox, because a fox suggests the word 
“fox” as well as vice versa. When, the fox having just disappeared, 
I utter the word “fox”, and when you, having not yet seen the 

129 1 

human knowledge: its scope and limits 

animal, hear the word, there is “belief” in the sense defined above. 
So there is when, without speaking, you look for the fox. But it 
is only when action is suspended that belief becomes a definite 
state of mind — for instance when you have just looked up a train 
that you mean to take to-morrow. When immediate action is called 
for, energy may be drained into the muscles, and “belief” may 
be shown as merely a characteristic of bodily movements. But it 
must be remembered that shouting “fox” or “tally-ho” is a bodily 
movement; we cannot therefore deny that bodily movements may 
express beliefs. 

External reference, which we discussed in the last chapter, 
exists in all indicative words when used in the way in which the 
use of words begins. It exists also in non-verbal behaviour, as 
when a dog scratches at a rabbit- hole because he has seen a rabbit 
go down the hole. But when behaviour is non-verbal it is difficult 
for the observer, and often for the agent, to say what, exactly, it 
is that the behaviour refers to. Words, like balances and thermo- 
meters, are instruments of precision, though often not very good 
ones; but that to which they give precision can exist, and be 
apprehended vaguely, without their help. 

To put the matter schematically, with a more or less unreal 
simplification : the presence of a stimulus A causes a certain kind 
of behaviour, say B; as a result of experience, something else, 
say C, may cause B in the absence of A. In that case, C may be 
said to cause “belief” in A, and “belief” in A may be said to be 
a feature of the behaviour B. When words come in, all this 
becomes more precise. The sight of a fox (A) causes you to 
pronounce the word “fox” (B); you may learn the trail of a fox 
in snow (C), and, seeing it, say “fox”. You are then “believing” 
A because of C. And if the trail was made by a fox, your belief 
is true. 

That which has external reference — the belief or idea or bodily 
movement — is in some cases public and in others private. It is 
public when it consists in overt behaviour, including speech; it 
is private when it consists of images or “thoughts”. (The meaning 
of “public” and “private” in this connection will be explained 
in Part III.) When an occurrence in an organism has external 
reference, the only feature always present is the causal one 
explained in the last paragraph, namely that the occurrence has 
some of the effects that would result from the sensible presence 


truth: elementary forms 

of that which is its external reference. We will give the name 
“representational occurrence” to anything that happens in an 
organism and has external reference. 

In addition to the essential causal relation by which “repre- 
sentational occurrence” is defined, further relations exist in certain 
kinds of such occurrences. In a memory image there is resemblance 
to what is represented (i.e. remembered). In other images there 
is likely also to be resemblance, though of a less exact kind. If 
somebody tells you “your son has been killed by falling over a 
precipice”, you are likely to have a very vivid image which will 
be correct in some respects but not in others. But words (except 
when onomatopoeic) have no resemblance to what they mean, 
and therefore verbal beliefs cannot be judged true or false by 
likeness to, or difference from, what they assert. Verbal behaviour 
is only one form of bodily behaviour that is representational; 
another form is that of the dog scratching at the rabbit-hole. We 
may say quite generally that bodily behaviour, when represen- 
tational, need not have any resemblance to that to which it refers. 

Nevertheless, in the case of explicit language, there is often 
a structural resemblance between a sentence and what it asserts. 
Suppose you see a fox eat a goose, and afterwards you say “the 
fox ate the goose”. The original occurrence was a relation between 
a fox and a goose, while the sentence creates a relation between 
the word “fox” and the word “goose”, namely that the word 
“ate” comes between them. (Cf. Wittgenstein’s Tractatus.) This 
possible structural similarity between a sentence and what it 
asserts has a certain importance, but not, I think, an importance 
which is fundamental. 

The above account of what makes a representational occurrence 
“true” is, I think, correct when it is applicable, but there are 
various extensions which give “truth” a wider scope. 

Let us begin with memory. You may recollect an event which 
calls for no present action, and in that case the above definition 
of “true” is not applicable. Your memory, if it is in images, may 
then be “true” in the sense of being like the event. And even if 
no present action is called for, there may be future situations in 
which your memory has practical importance, and it may now 
be called “true” if it will then fulfil the test. 

But what is of more importance is what may be called “deriva- 
tive” meaning, which is a property of sentences whose several 


human knowledge: its scope and limits 

words have “primary” meaning. Suppose that, for a given child, 
the words “cat”, “dog”, and “hate” have primary meaning, in 
the sense that they have been learnt by ostensive definition. Then 
the sentence “cats hate dogs” has a meaning which does not have 
to be learnt by a fresh process of either ostensive or nominal 
definition. It is, moreover, a sentence which can never be verified 
by one sensible occurrence; in this it differs from “there will be 
a loud noise in a moment”. Only in Plato’s heaven could we see 
THE CAT hating THE DOG. Here on earth, the facts in virtue 
of which the sentence is true are many, and cannot all be expe- 
rienced at one time. The relation of the sentence to the facts in 
virtue of which it is true is derivative from a number of other 
sentences, each of the form: “This is a cat and that is a dog, 
and this hates that.” (I am taking “hate” as a characteristic of 
overt behaviour; I am doing this not as a theory, but for purposes 
of illustration.) We have here three sentences, (a) “this is a cat”, 
(b) “that is a dog”, ( c ) “this hates that”. Each of these can be 
directly caused by the present sensible facts, given that the 
observer has learnt English. A sufficient number of such sen- 
tences, or of the corresponding observations or beliefs, will, in 
most people, in time cause the sentence “cats hate dogs”, of 
which the meaning follows by the laws of syntax from the meanings 
of sentences of the forms (a), (6), and (c). It is in this sense that 
the meaning of such sentences is “derivative”. For the present, 
having observed that the meaning of most sentences is derivative, 
I wish to confine myself to sentences of which the meaning is 

Let us consider the sentence “this is a cat”, uttered when a 
cat is sensibly present. Hitherto I have been considering “truth” 
and “falsehood” as ideas only applicable to representations of 
things not sensibly present, and if we adhered strictly to this view 
our sentences (a), ( b) y and (c), when uttered, would be neither true 
nor false. This way of using the words would, however, be incon- 
venient, and I should prefer to say that ( a ), (£), and (c) are true 
or false. 

If, in the presence of an animal, I say “this is a cat”, what are 
the possibilities of falsehood? There is, first, deliberate lying: I 
may be talking to a blind man, and wish him to think it a cat 
when it is really a rabbit. We may exclude this case, on the 
ground that the words I utter do not express a belief, and on the 


truth: elementary forms 

further ground that the word “cat” is not caused by what I see, 
but by some ulterior motive. Then there is the case where I do 
not see distinctly, owing to darkness or bad eyesight, and when 
some one turns on another light I say “oh, I see it was not a 
cat”. In this case what I see must have some likeness to a cat, 
and if I had said “this is something resembling a cat” I should 
have spoken truly. Then there is the case of insufficient know- 
ledge of the language, leading me to give the name “cat” to what 
is officially called a puma. In this case there is only social error: 
my language is not correct English, but in the language that I 
speak my statement is true. Finally, I may be suffering from 
delirium tremens and see a cat where there is nothing, at least 
from a public point of view. In the absence of such unusual 
possibilities, my statement “this is a cat” will be true. 

When there is a cat, and I say “there is a cat”, what is the 
relation of what I say to the actual present quadruped ? There 
is a causal relation: the sight of the cat causes the word “cat”, 
but this, as we saw in the case of indistinct vision, is not enough 
to insure truth, since something not a cat may cause the word 
“cat”. When I say “this is a cat”, I am asserting the existence 
of something which is not merely a momentary visual experience 
of my own, but lives and breathes and mews and purrs and is 
capable of feline joys and sorrows. All this is erroneous in the 
case of delirium tremens. Let us therefore take a simpler example, 
say “this is blue”. This statement need not imply anything beyond 
an experience private to me, and is therefore not liable to the 
kind of error that afflicts the drunkard. In this case, the only 
possibility of error is ignorance of the language, leading me to 
call “blue” what others would call “violet”. This is social error, 
not intellectual error; what I am believing is true, but my words 
are ill chosen. In this case, therefore, the possibility of genuine 
falsehood in my statement seems to be at a minimum. 

We may say generally: an indicative word is true when it is 
caused by what it means, assuming that the word is used in what 
may be called the exclamatory manner, as when people shout 
“fire I” or “murder!” In developed speech, we usually drop this 
way of using indicative words, and, instead, preface the word 
by “this is”. Thus the statement “this is blue” is true if it is 
caused by what “blue” means. This is in fact a tautology. But 
most words, such as “cat” and “dog”, mean not only what can 


HUMAN knowledge: its scope and limits 

be a momentary percept, but also the habitual concomitants of 
this kind of percept. If these are only usual but not invariable 
concomitants, there may be error in using the word that the 
percept causes; this is the case of the victim of delirium tremens , 
and also of Isaac when he mistook Jacob for Esau. Most words 
embody animal inductions which are usually true, but not always; 
this applies, more particularly, to names of objects or kinds of 
objects, such as our friends or the various species of animals. 
Whenever such words are employed as a result of a percept there 
is, therefore, some possibility of error, though often only a very 
slight one. 

We may now say, as a definition: A sentence of the form “this 
is A” is called “true” when it is caused by what “A” means. We 
may say further that a sentence of the form “that was A” or 
“there will be A” is “true” if “this is A” was, or in the second 
case will be, true in the above sense. This covers all sentences 
asserting what are, were, or will be, facts of perception, and also 
those in which, from a percept, we correctly infer its usual con- 
comitants by animal inference, in so far, at least, as such con- 
comitants can form part of the meaning of an indicative word. 
This covers all the factual premisses of empirical knowledge. It 
does not cover general statements, such as “dogs bark”, nor yet 
principles of inference, whether deductive or non- demonstrative. 
These cannot be adequately considered until we have dealt with 
the meaning of logical words such as “or” and “all”. The above, 
moreover, is only a definition of “truth”, not of “falsehood”. 
“Falsehood” remains to be dealt with later. 

There is one important observation to be made about our 
definitions of “meaning” and “truth”, and that is that both depend 
upon an interpretation of “cause” which, according to modern 
physics, might seem to be crude and only partially applicable 
to natural processes. If this view is adopted, it follows that what- 
ever defects belong to this old-fashioned notion of “cause” belong 
also to the notions of “meaning” and “truth” as we have inter- 
preted them. I do not think, however, that this is a very serious 
objection. Both concepts, on other grounds, are necessarily some- 
what vague and inexact, and these other grounds do much more 
to prevent precision than is done by modem physics. Such 
propositions as “lightning causes thunder”, “micro-organisms 
cause fevers”, “wounds cause pain”, although they have not the 


truth: elementary forms 

certainty formerly attributed to them, and even if (what for 
reasons that will appear later, I do not believe) “cause” is a 
rough-and-ready notion belonging to a certain stage of science, 
not a fundamental category as used to be thought, nevertheless 
express in a convenient form truths about the usual though not 
invariable course of nature, and as such are still useful except 
where, as in quantum physics, the last refinement of accuracy 
is sought in spite of its complication and its consequent useless- 
ness for most purposes of prediction. If human behaviour could 
be calculated by the physicists, we should have no need of such 
concepts as “meaning”, “belief”, and “truth”. But in the mean- 
time they remain useful, and up to a point they can be freed 
from ambiguity and vagueness. Beyond this point it would be 
useless to attempt to go, if “cause” is in fact not a fundamental 
concept of science. But if, as I hold, the concept of “cause” 
is indispensable, then the above considerations do not arise, or at 
any rate arise only in a modified form. 


Chapter IX 


I N the preceding Chapter we dealt with the truth of beliefs 
and sentences in cases where this depends only upon ob- 
servation and not upon inference from previous knowledge. 
In this Chapter we have to begin the inquiry into sentences 
of kinds that can be proved or disproved when suitable data 
derived from observation are known. Where such sentences 
are concerned, we no longer have to consider the relation of 
beliefs or sentences to something which is in general neither a 
belief nor a sentence; we have instead to consider only syntactical 
relations between sentences, in virtue of which the indubitable 
or probable truth or falsehood of a certain sentence follows from 
the truth or falsehood of certain others. 

In such inferences there are certain words, which I shall call 
“logical” words, of which one or more always occur. These words 
are of two kinds, which may be called respectively “conjunctions” 
and “general words”, though not quite in the usual grammatical 
sense. Examples of conjunctions are “not”, “or”, “and”, “if- 
then”. Examples of general words are “all” and “some”. (Why 
“some” is called a “general” word will appear as we proceed.) 

By the use of conjunctions we can make various simple in- 
ferences. If “p” is true, “not-p” is false; if “p” is false, “not-p” 
is true. If “p” is true, “p or q y is true; if “<7” is true, “p or <7” 
is true. If “p” is true and “<7” is true, “p and q” is true. And so 
on. Sentences containing conjunctions I shall call “molecular” 
sentences, the “p” and “<7” which are conjoined being conceived 
as the “atoms”. Given the truth or falsehood of a set of proposi- 
tions, the truth or falsehood of every molecular proposition con- 
structed out of the set follows by syntactical rules, and requires 
no fresh observation of facts. We are, in fact, in the domain of 

Given that we know about “p”, both what is involved in 
believing “p” and what would make “p” true or false, what can 
we say about “not -p”? 

Given a sentence “p”, we may either believe or disbelieve it. 
Neither of these is the primary use of a sentence; the primary 



use is to express belief in something else. If, feeling a drop on 
my nose, I say “it is raining’’, that is what may be called “primary” 
assertion, in which I pay no attention to the sentence, but use 
it to refer directly to something else, namely the rain. This kind 
of assertion has no corresponding negative. But if you say to me 
“is it raining?” and I then look out of the window, I may answer 
“yes” or “no”, and the two answers are, so to speak, at the same 
level. In this case I am presented first with a sentence, and after- 
wards, because of the sentence, with a meteorological fact which 
enables me to say “yes” or “no”. If I answer “yes”, I am not 
saying “it is raining”, but “the sentence ‘it is raining’ is true”; 
for what was presented to me by your question was a sentence, 
not a meteorological fact. If I answer “no”, I am saying “the 
sentence ‘it is raining’ is false”. This suggests that perhaps I could 
interpret “it is not raining” as meaning “the sentence ‘it is raining’ 
is false”. 

There are, however, two difficulties about such a view. The 
first is that it will make it very difficult to see what we mean 
by “false”; the second is that it makes it almost impossible to 
understand how a sentence containing the word “not” can be 
found true by observation. When, in answer to your question, 
I look out of the window, I do not merely not observe that it 
is raining, for I could have achieved this without looking out; 
in some sense I observe that it is not raining, but what this sense 
can be is obscure. 

How do I know what I assert when I say “it is not raining”? 
I may say: “I saw the whole sky was blue, and I know it does 
not rain when the sky is blue”. But how do I know this? Because 
I have often simultaneously observed facts which I could assert 
in the two sentences “the sky is blue” and “it is not raining”. 
So I cannot in this way explain how I come to know negative 

In what sense, if any, are there negative facts, as opposed to 
true sentences containing the words “not”? Let us put the matter 
as follows: Imagine a person who knew everything that can be 
stated without using the word “not” or some equivalent; would 
such a person know the whole course of nature, or would he not? 
He would know that a buttercup is yellow, but he would not 
know that it is not blue. We may say that the purpose of know- 
ledge is to describe the world, and that what makes a judgment 



of perception true (or false) is in general something that would 
still be a fact if there were no judgments in the world. The 
yellowness of the buttercup may be taken to be such a fact, and 
must be mentioned in a complete description of the world. But 
would there be the buttercup’s not-blueness if there were no 
judgments ? And must we, in a complete description of the butter- 
cup, mention all the colours that it is not? 

Let us consider a case where perception leads us as directly 
as possible to a very simple negative judgment. Suppose you take 
sugar thinking it is salt ; when you taste it you are likely to exclaim 
“this is not salt”. In such a case there is a clash between idea and 
sensation : you have the idea of the taste of salt, and the sensation 
of the taste of sugar, and a shock of surprise because the two are 
so different. Perception only gives rise to a negative judgment 
when the correlative positive judgment had already been made 
or considered. When you look for something lost, you say “no, 
it’s not there”; after a flash of lightning you may say “I have not 
heard the thunder”. If you saw an avenue of beeches with one 
elm among them, you might say “that’s not a beech”. If some 
one says the whole sky is blue, and you descry a cloud on the 
horizon, you may say “that is not blue”. All these are very obvious 
negative judgments resulting, fairly directly, from perception. Yet, 
if I see that a buttercup is yellow, I hardly seem to be adding 
to my knowledge by remarking that it is not blue and not red. 
What, then, is meant, in the way of objective fact, by a true 
negative judgment ? 1 

In all spontaneous negative perceptive judgments the experience 
which leads to the judgment is, in its essential core, of one and 
the same kind. There is an image or idea of a sensation of a certain 
sensational class, and there is a sensation of the same class but 
different from that of which there was an idea. I look for blue, 
and I see red; I expect the taste of salt, and I get the taste of 
sugar. Here everything is positive : idea of blue, sensation of red, 
experience of difference. When I say “difference” I do not mean 
mere logical non-identity, such as exists (e.g.) between a colour 

1 In what follows I am concerned to show that it is possible to define 
the truth of negative judgments without assuming that there are negative 
facts. I profess only to construct one theory which secures this result; 
I do not contend that there is no alternative theory which might be 
equally satisfactory. 



and a taste; I mean the sort of difference that is felt between 
two colours. This sort of difference is a matter of degree. We can 
pass from blue to red by a series of intermediate shades, each 
of which is subjectively indistinguishable from the next. We can 
say that between two shades of colour there is a “great” difference, 
which would be meaningless if said of a colour and a taste. Two 
shades of colour have a certain kind of incompatibility: when 
I see blue in a certain direction, I do not simultaneously see red 
in that direction. Other kinds of sensation have a similar incom- 
patibility ; at any rate this is true of sensations of touch : if I feel 
a given part of the body tickled, I do not simultaneously feel it 

When, as a result of perception, I say “this is not blue”, I may 
be interpreted as meaning “this is a colour differing from blue”, 
where “differing” is the positive relation that might be called 
“dissimilarity”, not abstract non-identity. At any rate, it may be 
taken that this is the fact in virtue of which my judgment is true. 
We have to distinguish between what a judgment expresses and 
what it states, i.e. what makes it true or false. Thus when I say 
truly “this is not blue”, there is, on the subjective side, con- 
sideration of “this is blue”, followed by rejection, while on the 
objective side there is some colour differing from blue. In this 
way, so far as colour judgments are concerned, we escape the 
need of negative facts as what make negative judgments true. 

But there remains a difficulty, and a very serious one. The 
above theory only succeeds in virtue of the incompatibility of 
different colours, i.e. of the fact that if I see red in a given 
direction I do not simultaneously see blue in that direction. This 
reintroduces “not”, which we were trying to get rid of. If I 
could see both blue and red simultaneously in a given direction, 
then “this is red” would not be a ground for “this is not blue”. 
The impossibility of seeing two colours simultaneously in a given 
direction feels like a logical impossibility, not like an induction 
from experience; but this is only one of various hypotheses that 
are prima fade possible. Suppose that, in a given direction from 
my eye, there were a source of red light, and also a source of blue 
light directly behind it; I should then have some colour sensation, 
which might not be either red or blue, but would be of some 
single shade of colour. It would seem that the different shades 
of colour are the only sensations of their kind that are physiolo- 


HUMAN knowledge: its scope and limits 

gically possible, and that there is nothing analogous to hearing 
a chord in music. 

Let us examine the hypothesis that the incompatibility of red 
and blue is logical, and ask ourselves whether this helps us in 
eliminating “not” from the objective world. We are now sup- 
posing it a tautology to say: “if there is red at a given moment 
in a given direction in the visual field, there is not blue in that 
direction at that time”. More simply, though less accurately, we 
may state our supposition as saying: “It is logically impossible 
that ‘this is red 1 and ‘this is blue’ should both be true of a given 
‘this’.” But this supposition, whether true or false, will not help 
us. Two positive predicates, as Leibniz pointed out in proving 
that God is possible, cannot be logically incompatible. Therefore 
our supposition requires us to regard either “red” or “blue” or 
both as complex, and one at least must contain a “not” in its 
definition. For, given two complex predicates P and Q, they will 
only be logically incompatible if one of them contains a con- 
stituent A and the other contains a constituent not-A. In this 
sense “healthy” and “ill” are incompatible, and so are “alive” 
and “dead”. But there can never be logical incompatibility except 
what is ultimately derived from the incompatibility of two pro- 
positions/) and not-/>. Therefore we cannot eliminate “not” from 
the objective world if we suppose red and blue to be logically 

Let us examine more carefully the view that the incompatibility 
of red and blue has a physiological source. That is to say, we are 
to suppose that a stimulus of a certain kind causes a sensation 
of red, while a stimulus of another kind causes a sensation of blue. 
I incline to think this the best theory, but we then have to explain 
the incompatibility of the two kinds of stimuli. As a matter of 
physics, this incompatibility may be taken to arise from the fact 
that each light-quantum has one definite amount of energy, 
together with the quantum laws connecting energy and frequency. 
The difficulty here is that it is not enough to say of a given light 
quantum that it has such-and-such an amount of energy; we 
must also be able to say that it does not have also some other 
amount. This is always regarded as so self-evident that it is never 
even stated. In classical physics analogous principles might have 
had a logical basis, but in quantum physics the incompatibility 
seems synthetic. 



Let us make a new start in the endeavour to eliminate negative 
facts. Given a single indicative sentence, such as “this is red”, 
we may have towards it two attitudes, belief and disbelief. Both 
are “positive” in the sense that they are actual states of the 
organism, which can be described without the word “not”. Each 
is capable of being “true”, but the “truth” of a disbelief is not 
quite the same thing as that of a belief. We considered in a 
previous Chapter what is meant by the “truth” of a perceptive 
belief: “this is red” is “true” if it is caused by something red. 
We did not then define what makes the corresponding disbelief 
“true”. Let us now address ourselves to this question. 

If disbelief in “this is red” is a judgment of perception — which 
is the case that we are considering — then “this” must be a colour. 
It is only in logic or philosophy that we are concerned to dis- 
believe in the redness of smells or sounds, and such disbelief 
belongs to a later stage than that which has to be considered in 
relation to our present problem. I shall therefore assume that when, 
as a judgment of perception, we disbelieve “this is red”, we are 
always perceiving that it is some other colour. We may therefore 
say that a disbelief in “this is red” is “true” when it is caused 
by something having to red the relation of positive dissimilarity 
which we considered earlier. (This is a sufficient, not a necessary, 

We must now interpret the law of contradiction. We must not 
say “ ‘this is red’ and ‘this is not red’ cannot both be true”, since 
we are concerned to eliminate “not”. We must say “A disbelief 
in the sentence ‘the belief that this is red and the disbelief that 
this is red are both true' is always true”. It seems that in this 
way we can replace “not” and “falsehood” by “disbelief” and 
“the truth of a disbelief”. We then reintroduce “not” and “false- 
hood” by definitions: the words “this is not blue” are defined 
as expressing disbelief in what is expressed by the words “this 
is blue”. In this way the need of “not” as an indefinable con- 
stituent of facts is avoided. 

The above theory may be summarized as follows: As a matter 
of logic, if any propositions containing the word “not” are known, 
there must be, among uninferred propositions, some that are of 
the form “not -/>” or of the form “ p implies not-j”. It seems that 
a judgment “this is not red” may be a judgment of perception, 
provided “this” is a colour other than red. The judgment may 


HUMAN knowledge: its scope and limits 

be interpreted as disbelief in “this is red”, disbelief being a state 
just as positive as belief. A sufficient (not necessary) condition 
for the truth of disbelief in “this is red” is that the disbelief 
should be caused by a “this” having to red the relation of positive 

There is another sufficient, not necessary, test of truth in certain 
cases. “This is blue” is “true” if a person whose belief is expressed 
by these words will, in suitable circumstances, have a “quite-so” 
feeling, and is “false” if he will get a “how-surprising” feeling. 
To every belief there is a corresponding disbelief. A person 
“disbelieves” what is expressed by “this is blue” if he will be 
surprised if “this is blue” is true, and have a “quite-so” feeling 
if “this is blue” is false. The words “this is not blue” (to repeat) 
express disbelief in what is expressed by “this is blue”. Speaking 
generally, “not-p” must be defined by what it expresses . 

The purpose of this theory is to explain how negative sentences 
can be true, and can be known, without its being necessary to 
assume that there are facts which can only be asserted in sentences 
containing the word “not”. 

All empirical negative judgments are derived from negative 
judgments of perception of the type of “this is not blue”. Suppose 
you see an animal at a distance, which at first you take to be a 
dog, but which, on a nearer approach, turns out to be a fox. This 
depends upon perception of shape, and perception of shape 
depends upon the fact that where you see one colour you do not 
see another. The moment when you say “that is not a dog, but 
a fox” is the moment when you see something that you did not 
expect, say the fox’s brush. When your surprise is analysed it 
comes down to some such judgment of perception as “this is not 
green but brown”, where the fox’s brush unexpectedly hides the 

There is more to be said about negation, in connection with 
general propositions, and also with logic. But the above analysis 
seems adequate where negative judgments of perception are con- 
cerned, and generally in all cases in which observation leads us 
to assert a sentence containing the word “not”. 

We must now attempt a similar treatment of the word “or”. 

In the case of “or” it is even more obvious than in the case 
of “not” that what makes “ p or q ” true is not a fact containing 
some constituent corresponding to “or”. Suppose I see an animal 



and say “that was a stoat or a weasel”. My statement is true if 
it was a stoat, and true if it was a weasel; there is not a third 
kind of animal, stoat-or- weasel. In fact, my statement expresses 
partial knowledge combined with hesitation; the word “or” 
expresses my hesitation, not something objective. 

But it is possible to raise objections to this view. It may be said 
that the word “stoat” denotes a class of animals, not all exactly 
alike, and that the same is true of the word “weasel”. The phrase 
“stoat Or weasel”, it may be said, merely denotes another class 
of animals, which, like each of the previous classes, is composed 
of individuals having common characteristics combined with 
differences. There might easily be one word for stoat-or- weasel, 
say “stosel”, and we could then say “that was a stosel”. This 
would assert, without “or”, the same fact previously asserted with 
that word. 

Or, to take a simpler instance : there are many shades of blue, 
having different names; there is navy-blue, aquamarine, peacock- 
blue, and so on. Suppose we have a set of shades of blue, which 
we will call b lt b 2) and so on, and suppose everything blue is of 
one of these shades. Then the statement “this is b v or b 2 , or etc.”, 
is precisely equivalent to “this is blue”, but the first statement 
contains “or” while the other does not. 

Such facts, however, rightly interpreted, confirm the view that 
the meaning of “or” is subjective. The word “or” can be elimi- 
nated without making any difference to the fact that makes a 
sentence true or false, but not without making a difference to 
the state of mind of a person asserting the sentence. When I say 
“that is a stoat or a weasel”, I may be supposed to add “but I 
don’t know which”; when I say “that is a stosel”, this addition 
is absent, though it might still be true if I made it. In fact “or” 
expresses conscious partial ignorance, although in logic it is 
capable of other uses. 

There is in this respect a difference between the standpoint 
of logic and that of psychology. In logic, we are only interested 
in what makes a sentence true or false ; in psychology, we are also 
interested in the state of mind of the person uttering the sentence 
with belief. In logic, “/>” implies “p or q” , but in psychology the 
state of mind of a person asserting “p” is different from' that of 
a person asserting “p or q'\ unless the person concerned is a 
logician. Suppose I am asked “what day was it you went to 


HUMAN knowledge: its scope and limits 

London?” I may reply “Tuesday or Wednesday, but I don’t 
remember which”. If I know that it was Tuesday, I shall not 
reply “Tuesday or Wednesday”, in spite of the fact that this 
answer would be true. In fact we only employ the word “or” 
when we are uncertain, and if we were omniscient we should 
express our knowledge without the use of this word — except, 
indeed, our knowledge as to the state of mind of those aware 
of a greater or less degree of ignorance. 

The elimination of disjunctive “facts” is not so difficult as the 
elimination of negative “facts”. It is obvious that, although I may 
believe truly that to-day is Tuesday or Wednesday, there is not, 
in addition to Tuesday and Wednesday, another day of the week, 
called “Tuesday-or-Wednesday”. What I believe is true because 
to-day is Tuesday, or because to-day is Wednesday. Here “or” 
appears again, and it is true that we cannot define “or”. But 
what we cannot define is not a characteristic of the non-cognitive 
world, but a form of partial cognition. 

Someone might argue: “When I believe ‘p or q 9 I am clearly 
believing something , and this something is neither *p* nor *q\ 
therefore there must be something objective which is w r hat I am 
believing”. This argument would be fallacious. We decided that 
when I am said to be believing “not -/>” I am really disbelieving 
that is to say, there is a sentence not containing the word 
“not”, which denotes a certain content that I may believe or 
disbelieve, but when the word “not” is added the sentence no 
longer expresses merely a content, but also my attitude towards 
it. The case of “or” is closely analogous. If I assert “to-day is 
Tuesday or Wednesday”, there are two sentences, “to-day is 
Tuesday” and “to-day is Wednesday”, each of which denotes a 
certain content. My disjunctive assertion expresses a state of mind 
in which neither of these contents is either affirmed or denied, 
but there is hesitation between the two. The word “or” makes 
the sentence one which no longer denotes a single content, but 
expresses a state of mind towards two contents. 

When an indicative sentence is asserted, there are three things 
concerned. There is the cognitive attitude of the assertor — belief, 
disbelief and hesitation, in the cases so far considered; there is 
the content or contents denoted by the sentence ; and there is the 
fact or facts in virtue of w r hich the sentence is true or false, which 
I will call the “verifier” or “falsifier” of the sentence. In the 



sentence “To-day is Tuesday or Wednesday”, the cognitive atti- 
tude is hesitation, the contents are two, namely the significations 
of “to-day is Tuesday” and “to-day is Wednesday”; the verifier 
may be the fact that it is Tuesday or the fact that it is Wednesday, 
or the falsifier may be that it is a different day of the week. 

A sentence containing no logical words can only express belief. 
If we knew all true sentences containing no logical words, and 
also knew that they were all, every other true sentence could be 
obtained by logical inference. A sentence not in the list would 
become true by insertion of the word “not”. A sentence in which 
two sentences are connected by the word “or” would be true if 
either component sentence occurred in the list. A sentence in 
which two sentences are connected by the word “and” would be 
true if both component sentences occurred in the list. The same 
kind of logical proof would be possible for sentences containing 
the logical words “all” and “some”, as will be shown in the next 

Thus if we give the name “atomic sentence” to one not con- 
taining logical words, we should need, as premisses for omniscience 
(a) a list of all true atomic sentences, (6) the sentence “all true 
atomic sentences occur in the above list”. We could then obtain 
all other true sentences by logical inference. 

But the above method fails without ( b ), when we wish to estab- 
lish the truth of a sentence containing the word “all” or the 
falsehood of a sentence containing the word “some”. We can, 
no doubt, find substitutes for ( b ) but they will all contain, as it 
does, the word “all”. It seems to follow that our knowledge must 
embrace premisses containing this word, or, what is equivalent, 
asserting the falsehood of sentences containing the word “some”. 
This brings us to the explicit consideration of the words “all” 
and “some”, which will be the subject of the next Chapter. 


Chapter X 


B y “general knowledge” I mean knowledge of the truth or 
falsehood of sentences containing the word “all” or the 
word “some” or logical equivalents of these words. The 
word “some” might be thought to involve less generality than 
the word “all”, but this would be a mistake. This appears from 
the fact that the negation of a some-sentence is an all-sentence, 
and vice-versa. The negation of “some men are immortal” is 
“all men are mortal”, and the negation of “all men are mortal” 
is “some men are immortal”. Thus any person who disbelieves 
a some-sentence must believe an all-sentence, and vice versa. 

The same element of universality in a some-sentence appears 
from a consideration of its meaning. Suppose I say “I met a 
negro in the lane”. My statement is true if I met any member 
of the whole class of regroes; thus the whole class is relevant, 
just as much as it would be if I said “all negroes are of African 
descent”. Suppose you wanted to disprove my statement, there 
would be two things you could do. First, you could go through 
the whole class of negroes and prove that none of them were in 
the lane; secondly, you could go through the class of people I 
met, and prove that none of them were black. In either case a 
complete enumeration of some class is necessary. 

But as a rule a class cannot be completely enumerated. No 
one can enumerate the class of negroes. If it is to be possible to 
enumerate the class of people I met in the lane, we must be 
able to know, concerning any member of the human race, whether 
or not I met him in the lane. If I know, on a basis of perception, 
that I met A, B, and C, and no one else, then I must be supposed 
to know the general proposition “all human beings other than 
A, B, and C were not met by me”. This raises in an acute form 
the question of negative judgments of perception which we 
considered in the preceding Chapter. It also makes it evident 
that there are difficulties in disproving some-sentences, and 
correlatively in proving all-sentences. 

But before considering further the truth or falsehood of such 
sentences, let us first examine what they signify. 



It is clear that the sentence “all men are mortal” can be 
understood by a person who is unable to give a list of all human 
beings. If you understand the logical words involved, and also 
the predicates “man” and “mortal”, you can fully understand 
the sentence, whether or not you can know its truth. Sometimes 
you can quite certainly know the truth of such a sentence although 
enumeration of the class concerned is impossible; an example 
is “all primes other than 2 are odd”. This of course is a tautology; 
so is the statement “all widows have been married”, which is 
not known by means of an enumeration of widows. In order to 
understand a general sentence, only intensions need be under- 
stood; the cases in which extensions are known are exceptional. 

Further: when an intension is first given, enumeration of the 
corresponding extension is only possible through a universal 
negative. Given, e.g. that A, B, C . . . inhabit a certain village, 
this only gives the extension of “inhabitant of this village” if 
we know “no human being except A, B, C . . . inhabits this 
village”. Thus unless a class is defined by enumeration, it can 
only be enumerated by the help of some negative all-sentence 
which must be supposed known. 

Although, in pure logic, an all-proposition cannot be proved 
except by means of premisses which are all-propositions, there 
are many all-propositions which we all believe for reasons derived 
from observation. Such are “dogs bark”, “men are mortal”, 
“copper conducts electricity”. The conventional view is that 
such propositions are inductive generalizations, which are prob- 
able, but not certain, when their premisses are known. We are 
supposed to know from observation “A is a dog and A barks”, 
“B is a dog and B barks”, and so on; we are also supposed not 
to know any proposition of the form “X is a dog and X does not 
bark”. It is supposed to follow that probably all dogs bark. I 
am not concerned at the moment with the validity of such infer- 
ences, but only with the fact that knowledge of the principle 
guaranteeing their validity, if it exists, is general knowledge, 
and of a sort which cannot be based on observation. Induction, 
therefore, even if valid, does not help us to understand how we 
come by general knowledge. 

There are three chief methods of arriving at general pro- 
positions. Sometimes they are tautologies, such as “all widows 
are female”; sometimes they result from induction; sometimes 


HUMAN knowledge: its scope and limits 

they are proved by complete enumeration, e.g. “everybody in 
this room is male”. I shall begin by considering complete 

From the point of view of knowledge, though not of logic, 
there is an important difference between positive and negative 
general propositions, namely that some general negative pro- 
positions seem to result from observation as directly as “this is 
not blue”, which we considered in the preceding Chapter. In 
Through the Looking-Glass the king says to Alice “who do 
you see coming along the road?”, and she replies “I see nobody 
coming”, to which the king retorts: “What good eyes you must 
have I It’s as much as I can do to see somebody by this light”. 
The point, for us, is that “I see nobody” is not equivalent to 
“I do not see somebody”. The latter statement is true if my 
eyes are shut, and affords no evidence that there is not somebody ; 
but when I say “I see nobody”, I mean “I see, but I do not see 
somebody”, which is prima facie evidence that there is not some- 
body. Such negative judgments are just as important as positive 
judgments in building up our empirical knowledge. 

Consider, for example, such a statement as “this village has 
623 inhabitants”. Census officials make such statements confi- 
dently on a basis of enumeration. But enumeration involves 
not only 623 propositions of the form “this is a human being”, 
but also an indefinite number of propositions of the form “this 
is not a human being”, and finally some assurance that I have 
enough such propositions to be fairly sure that no one has been 
overlooked. Jenghiz Khan believed the proposition “all the 
inhabitants of Merv have been killed”, but he was wrong, because 
some had crept into hiding-places that he overlooked. This was 
an actual source of error; another possible source would have 
been if some grotesque and long-immured prisoner had been 
wrongly judged by him to be a gorilla. 

Suppose you are a Gestapo officer engaged in a search, and 
you satisfy yourself that at a certain time a certain house contained 
just five people. What is involved in arriving at this knowledge? 
Whenever you perceive a human being in any part of the house, 
you cause him to come to a certain room ; when you are satisfied 
that none are left, you count those whom you can see, and find 
that there are five of them. This requires that, in the first place, 
you should have a number of judgments “I see a man in this 



direction* * and “I see in that direction something which is not 
a man**. It requires in the second place the judgment “in the 
process that I went through, any man in the house would have 
been perceived”. This second judgment is very likely to be mis- 
taken for common-sense reasons, and we may ignore it, but the 
other requires examination. 

When you answer “no** to such questions as “is there a man 
there?** “do you hear a noise?*’ “does that hurt?” you are 
asserting a universal negative, and yet your answer seems to 
result as directly from perception as if you had answered “yes’*. 
This must depend upon the kind of incompatibility discussed 
in the previous Chapter. You are seeing something, but its shape 
differs from that of a human being; your auditory consciousness 
is in the state of listening, but not of hearing; in the part of the 
body concerned you are feeling something other than pain. It 
is only in virtue of incompatibility that a positive percept gives 
rise to a universal negative: where I see blue, I can assert that 
I see no shade of red, provided the area involved is sufficiently 
small. Such universal negatives based on perception raise great 
difficulties, but without them most of our empirical knowledge 
would be impossible, including, as we have seen, everything 
statistical and everything arrived at by enumerating the members 
of a class defined by intension, such as “the inhabitants of this 
village” or “the people now in this room”. We must therefore 
somehow find a place in our theory of knowledge for universal 
negatives based on perception. 

I will, however, for the moment, leave this problem aside to 
examine whether there are general facts y as opposed to true 
general propositions; and, if general facts are rejected, what it 
is that makes general propositions true, when they are true. If 
this question has been decided, it may become easier to discover 
how true general propositions come to be known. 

Are there general facts ? We may re-state this question in 
the following form: Suppose I knew the truth or falsehood of 
every sentence not containing the word “all” or the word “some” 
or an equivalent of either of these words; what, then, should I 
not know? Would what I should not know be only something 
about my knowledge and belief, or would it be something that 
involves no reference to knowledge or belief? I am supposing 
that I can say “Brown is here”, “Jones is here”, “Robinson is 


human knowledge: its scope and limits 

here”, but not “some men are here”, still less “exactly three 
men are here” or “every man here is called ‘Brown* or ‘Jones* or 
‘Robinson* And 1 am supposing that, though I know the truth 
or falsehood of every sentence of a certain sort, I do not know 
that my knowledge has this completeness. If I knew my list 
to be complete I could infer that there are three men here, but, 
as it is, I do not know that there are no others. 

Let us try to make clear exactly what is involved. When the 
Antarctic Continent was discovered, something became known 
which had been there before anybody knew it; the knowing was 
a relation between a percipient and something which was 
independent of perception and generally of the existence of life. 
Is there anything analogous in the case of true all-sentences 
and some-sentences, e.g. “there are volcanoes in the Antarctic”? 

Let us give the name “first-order omniscience’* to knowledge 
of the truth or falsehood of every sentence not containing general 
words. “Limited first-order omniscience” will mean similar 
complete knowledge concerning all sentences of a certain form, 
say the form “ x is human”. We are to inquire what is not known 
to a person with first-order omniscience. 

Can we say that the only thing he does not know is that his 
knowledge has first-order completeness? If so, this is a fact 
about his knowledge, not about facts independent of knowledge. 
It might be said that he knows everything except that there is 
nothing more to know; it would seem that no fact independent 
of knowing is unknown to him. 

Let us take a case of limited first-order omniscience. Consider 
sentences of the form “ x is human” and “ x is mortal”, and let 
us suppose that a certain wise man knows whether these sentences 
are true or false, for every value of for which the sentences 
are significant, but does not know (what is in fact true) that there 
are no other values of ‘ V* for which the sentences are significant. 
Suppose A, B, C . . . Z are the values of ‘V* for which is 
human” is true, and suppose that for each of these values “ x is 
mortal” is true. Then the statements “A is mortal”, “B is mortal” 
. . . “Z is mortal”, taken together, are in fact equivalent to “all 
men are mortal”, that is to say, if one is true so is the other, and 
vice versa. But our wise man cannot know this equivalence. 
In any case, the equivalence involves the conjunction of “A is 
mortal”, “B is mortal” . . . “Z is mortal”, that is to say, it 



involves a sentence built up by repeated use of the word “and”, 
which is to be interpreted on the same lines as the word “or”. 

The relation of “and” and “or” is peculiar. When I assert 
“ p and #”, I can be regarded as asserting “/>” and asserting 
“#”, so that the “and” of “/> and q ” seems unnecessary. But if 
I deny “ p and #”, I am asserting “not -/) or not-#”, so that “or” 
seems necessary for interpreting the falsehood of a conjunction. 
Conversely, when I deny “/> or #”, I am asserting “not-/) and 
not-#”, so that conjunction is needed to interpret the falsehood 
of disjunction. Thus “and” and “or” are interdependent; either 
can be defined in terms of the other plus “not”. In fact, “and”, 
“or” and “not” can all be defined in terms of “not-/) or not-#”, 
and also in terms of “not-/) and not-#”. 

It is obvious that all-sentences are analogous to conjunctions, 
and some-sentences to disjunctions. 

Continuing with “all men are mortal”, let us allow our wise 
man to understand “and” and “or” and “not”, but let us still 
suppose him incapable of “some” and “all”. Let us further 
suppose, as before, that A, B, C . . . Z are all the men there 
are, and that our wise man knows “A is mortal and B is mortal 
and . . . and Z is mortal”; but since he does not know the 
word “all”, he does not know “A, B, C . . . Z are all the men 
there are”. Let us call this proposition “P”. The question that 
concerns us is : what, precisely, does he not know in not knowing P ? 

In mathematical logic, P is interpreted as: “Whatever x may 
be, either x is not human or x is A or x is B or . . . x is Z”. 
Or it may be interpreted as : “Whatever x may be, the conjunction 
‘ x is human and x is not A and x is not B and ... x is not Z’ 
is false.” Either of these is a statement about everything in the 
universe, and it seems preposterous to suppose that we can know 
about all the things in the universe. In the case of “all men” 
there is real doubt, since there may be men on a planet of some 
other star. But how about “all the men in this room”? 

We will now suppose that A, B, C are all the men in this room, 
that I know “A is in the room”, “B is in the room”, “C is in the 
room”, that I understand “and” and “or” and “not”, but not 
“all” or “some”, so that I cannot know “A and B and C are all 
the men in this room”. W 7 e will call this proposition “Q”. What 
do I not know in not knowing Q ? 

Mathematical logic still brings in everything in the universe 

15 1 


in interpreting Q, which it enunciates in the form: “Whatever x 
may be, either x is not in the room or x is not human or x is A 
or x is B or x is C”; or “Whatever x may be, if x is not A and 
x is not B and x is not C, then x is not human or x is not in the 
room”. But in this case the logistical interpretation, however 
convenient technically, seems obviously preposterous psychologic- 
ally, for in order to know who is in the room I obviously need 
not know anything about what is outside the room. How, then, 
is Q to be interpreted ? 

In practice, if I have seen A and B and C, and wish to be 
sure of Q, I look in cupboards, under tables, and behind curtains, 
and from time to time I say “there is no one in this part of the 
room”. Theoretically, I could divide the volume of the room 
into a number of smaller volumes, each just large enough to 
contain a small human being; I could examine each volume, 
and say “no one here” except where I found A and B and C. 
In the end, we must be able to say “I have examined all parts of 
this room” if we are to be justified in asserting Q. 

The statement “no one here” is analogous to “this is not blue”, 
which we considered in the preceding Chapter. It is not an indefi- 
nitely extended conjunction: “Brown is not here and Jones is not 
here and Robinson is not here and . . .”, through a catalogue of 
the human race. What it does is to deny a character which is 
common to places where there are human beings, and which we 
assert when we say “some one is here”, say in playing hide-and- 
seek. This raises no new problem. The universal is now in “I 
have examined all parts of the room” or some equivalent. 

The universal that we require may be stated as follows: “If I 
go through a certain process, every person in the room will become 
perceptible at some stage of the process.” The process must be 
one that can be actually carried out ; we should never be justified 
in saying “there are just three uranium atoms in this room”, but 
human beings, fortunately, are never microscopic. Our universal 
may be put in the form: “If I perform a certain series of acts 
Ax, A 2 , . . . A n , every human being within a certain volume V will 
be perceived during at least one of these acts.” This involves an 
almost inextricable tangle of logical, physical, metaphysical, and 
psychological elements, and as we are concerned at the moment 
only with the logical elements it will be better to choose another 
example to begin with. 



Let us take: “I have just heard six pips on the wireless.” This 
may be interpreted as: “During a brief recent period of time, I 
had exactly six closely similar auditory sensations of a certain well- 
defined sort, namely the sort called ‘pips’.” I may give proper 
names to each of these, say P 1# P 2 . . . . P 6 . Then I say: “P^ and 
P 2 and . . . and P 6 were all the pips I heard in the period from the 
time to the time J 2 .” We will call this statement “R”. 

It is fairly obvious that what distinguishes R from the con- 
junction “I heard Pj and I heard P 2 and . . . and I heard P 0 ” is 
negative: it is the knowledge that I heard no other pips. Let us 
consider this. Suppose I agree to listen for pips throughout a 
period of five seconds, at the beginning and end of which you say 
“now”. Immediately afterwards, you say “did you hear any pips”? 
and I say “no”. This, though logically a universal, may be psycho- 
logically a single negative judgment of perception, like “I don’t 
see any blue sky” or “I don’t feel any rain”. In such judgments 
(to repeat) we have the suggested idea of a quality and the sensa- 
tion of a different quality which causes us to disbelieve the sugges- 
ted idea. There is here no multiplicity of instances, but a specious 
present in which one quality is present and another is felt to be 
absent. We know “I did not hear pippiness” and we translate this 
into “I heard no pips”. The plurality of “pips” is that of events 
as opposed to qualities — a subject considered above in connection 
with proper names. 

We can extend such negative judgments beyond the specious 
present, because there is no sharp boundary between sensation 
and immediate memory, or between immediate memory and true 
memory. You say “do you hear a pip”, and I reply, not by a sharp 
“no”, but by a long-drawn out “no-o-o-o”. In this way my 
negation can apply to a period of ten seconds or so. By immediate 
memory and true memory it can have its temporal scope extended 
indefinitely, so as to justify such a statement as “I watched all 
night without seeing a single aeroplane”. W T hen such statements 
are legitimate, we can say “between the time t l and the time t 
I saw exactly six planes”, because we can divide the period into 
smaller ones, in six of which we say “I saw a plane”, and in the 
others we say “I saw no plane”. These various judgments are then 
assembled in memory, and give rise to the enumerative judgment 
“in the whole period I saw just six planes”. 

If the above theory is correct, negative judgments of perception 


HUMAN knowledge: its scope and limits 

are not themselves universal: they say (e.g.) “I did not hear 
pippiness”, not “I heard no pips”. The judgment “I heard no 
pips” follows logically, for a pip is a complex of which pippiness 
is a constituent. The inference is like that from “I saw no one” 
to “I saw no processions”. A procession is a crowd of human 
beings, and one man may at different times form part of many 
processions, but processions cannot exist without human beings. 
We can, therefore, from absence of the quality called “humanity”, 
logically infer the absence of processions. In like manner, from 
absence of noisiness we can infer absence of noises. 

If the above theory is correct, enumerative empirical judgments 
depend upon universal negative judgments logically inferable from 
negative perceptive judgments concerned with single qualities, 
such as “I do not see blue”. Our problem, so far as such judgments 
are concerned, is therefore solved by the preceding theories as to 
“not” and as to proper names. 

The above, however, is only one of the ways in which we arrive 
at general propositions. It is the way that is appropriate when 
complete enumeration is possible, i.e. when there is some property 
P of which we can say: “a ly a 2 , . . . a n are all the subjects of which 
P can be truly asserted”. It is applicable in arriving at “this 
village has 323 inhabitants”, or “all the inhabitants of this village 
are called Jones” or “all mathematical logicians whose names 
begin with Q live in the United States”. What we have been 
discussing is: “what is involved in the possibility of complete 
enumeration” 5 But there are multitudes of general propositions 
in which we all believe although complete enumeration is either 
practically or theoretically impossible. These are of two kinds, 
tautologies and inductions. Of the former kind are “all pentagons 
are polygons”, “all widows have had husbands”, etc. Of the latter 
kind are “all men are mortal”, “all copper conducts electricity”, 
etc. Something must be said about each of these kinds. 

Tautologies are primarily relations between properties, not 
between the things that have the properties. Pentagonality is a 
property of which polygonality is a constituent; it may be defined 
as polygonality plus quintuplicity. Thus whoever asserts penta- 
gonality necessarily asserts polygonality at the same time. Simi- 
larly “ x is a widow” means “ x had a husband who is dead”, and 
therefore asserts, incidentally, “ x had a husband”. We have seen 
that an element of tautology comes in when we seek to interpret 



such judgments as “I have heard no pips”. The strictly empirical 
element is “I have not heard pippiness”; “pips” are defined as 
“complexes of which pippiness is a constituent”. The inference 
from “not pippiness” to “no pips” is thus tautological. I shall 
say no more about tautological general propositions, since the 
subject belongs to logic, with which we are not concerned. 

It remains to consider inductive generalizations — not their 
justification, but their significance, and what facts are necessary 
if they are to be true. 

That all men are mortal could, theoretically, be proved by the 
enumerative method: some world-governing Caligula, having 
made a complete census, might extirpate his subjects and then 
commit suicide, exclaiming with his last breath: “Now I know 
that all men are mortal”. But in the meantime we have to rely 
upon less conclusive evidence. The most important question is 
whether such generalizations, when not proved by complete 
enumeration, are to be regarded as asserting a relation of intensions, 
whether certain or probable, or only a relation of extensions. And 
further : where there is a relation of intensions such as to justify 
“all A is B”, must this be a logical relation making the generaliza- 
tion tautological, or is there an extra-logical relation of intensions, 
of which we acquire probable knowledge by induction ? 

Take “copper conducts electricity”. This generalization was 
arrived at inductively, and the induction consisted of two parts. 
On the one hand, there were experiments with different bits of 
copper; on the other hand, there were experiments with a variety 
of substances, showing that, in every case that had been tested, 
each element has a characteristic behaviour as regards the con- 
duction of electricity. The same two stages exist in establishing 
the induction “dogs bark”. On the one hand, we hear a number of 
dogs barking; on the other hand, we observe that each species of 
animal, if it makes a noise at all, makes a noise characteristic of 
the species. But there is a further stage. The copper atom has been 
found to have a certain structure, and from this structure, to- 
gether with the general laws of physics, the conduction of electricity 
can be inferred. If we now define copper as “what has a certain 
atomic structure”, there is a relation between the intension 
“copper” and the intension “conductivity”, which becomes 
logical if the laws of physics are assumed. There is now, however, 
a concealed induction, namely that what appears as copper by 


human knowledge: its scope and limits 

the tests that were applied before the modern theory of atomic 
structure is also copper by the new definition. (This need only be 
true in general, not universally.) This induction itself could, 
theoretically, be replaced by deductions from the laws of physics. 
The laws of physics themselves are partly tautologies, but in their 
most important parts they are hypotheses that are found to explain 
great numbers of subordinate inductions. 

The same sort of thing may be said about “dogs bark”. From 
the anatomy of a dog’s throat, as from that of any musical wind 
instrument, it should be possible to infer that only certain sorts 
of sounds can issue from it. We thus replace the rather narrow 
inductive evidence derived from listening to dogs by the much 
wider evidence upon which the theory of sound depends. 

In all such cases the principle is the same. It is this: Given a 
mass of phenomena, everything about them except an initial 
space-time distribution follows tautologically from a small number 
of general principles, which we therefore take to be true. 

We are concerned at present, not with the validity of the 
grounds for these general principles, but with the character of 
what they assert, i.e. whether they assert relations of intension 
or purely extensional relations of class-inclusion. I think we must 
decide in favour of the former interpretation. When an induction 
seems plausible, that is because a relation between the intensions 
involved strikes us as not unlikely. “Logicians whose names 
begin with Q live in the United States” may be proved by com- 
plete enumeration, but will not be believed on inductive grounds, 
because we can see no reason why a Frenchman named (say) 
Quetelet should abandon his native country as soon as he became 
interested in logic. On the other hand, “dogs bark” is readily 
accepted on inductive grounds, because we expect a possible 
answer to the question “what sort of noise do dogs make?” What 
induction does, in suitable cases, is to make a relation of intensions 
probable. It may do this even in cases where the general principle 
suggested by induction turns out to be a tautology. You may 
notice that i + 3 = 2 2 , i + 3 + S = 3 2 , 1 + 3 + 5 + 7 = 4 2 > 
and be led to conjecture that the sum of the first n odd numbers 
is always n 2 ; when you have framed this hypothesis, it is easy to 
prove it deductively. How far ordinary scientific inductions, such 
as “copper conducts electricity”, can be reduced to tautologies, 
is a very difficult question, and a very ambiguous one. There are 



various possible definitions of “copper”, and the answer may 
depend upon which of these definitions we adopt. I do not think, 
however, that relations between intensions, such as justify state- 
ments of the form “all A is B”, can always be reduced to tautolo- 
gies. I am inclined to believe that there are such intensional 
relations that are only discoverable empirically, and that are not, 
either practically or theoretically, capable of logical demonstration. 

It is necessary, before leaving this subject, to say something 
about some-propositions, or existence-propositions as they are 
called in logic. The statement “some A is B” is the negation of 
“all A is not B” (i.e. “no A is B”), and “all A is B” is the negation 
of “some A is not B”. Thus the truth of some-sentences is equiva- 
lent to the falsehood of related all-sentences, and vice versa. We 
have considered the truth of all-sentences, and what we have said 
applies to the falsehood of some-sentences. Now we wish to con- 
sider the truth of some-sentences, which involves the falsehood 
of correlative all-sentences. 

Suppose I met Mr. Jones, and I say to you “I met a man”. This 
is a some-sentence: it asserts that, for some value of x , “I met x 
and x is human” is true. I know that the x in question is Mr. 
Jones, but you do not. What I know enables me to infer the truth 
of “I met a man”. There is here a distinction of some importance. 
If I know that the sentences “I met Jones” and “Jones is a man” 
are true, it is a substantial inference that the sentence “I met a 
man” is true. But if I know that I met Jones, and also that Jones 
is a man, then I am already knowing that I met a man. To know 
that the sentence “I met Jones” is true is not the same thing as to 
know that I met Jones. I can know the latter, but not the former, 
if I do not know English; I can know the former, but not the 
latter, if I hear it pronounced by a person for whose moral 
character I have the highest respect, but again I know no English. 

Suppose you hear the door- bell, and you infer that there is a 
caller. While you do not know who it is, you are in a certain state 
of mind, in which belief and uncertainty are combined. When 
you find out who it is, the element of uncertainty disappears, but 
the element of belief remains, together with the new belief “it is 
Jones”. Thus the inference from “a has the property P” to 
“something has the property P” consists merely in isolating and 
attending to a portion of the total belief expressed in asserting 
“a has the property P”. I think something of the same sort may 


HUMAN knowledge: its scope and limits 

be said about all deductive inference, and that the difficulty of 
such inference, when it exists, is due to the fact that we are be- 
lieving that a sentence is true rather than what the sentence asserts. 

The transition from sentences expressing judgments of per- 
ception to some-sentences, e.g. from “there’s Jones” to “there’s 
somebody”, thus offers no difficulty. But there are a number of 
some- sentences in which we all believe, but which are not arrived 
at in this simple way. We often know that something has the pro- 
perty P, although there is no definite thing a of which we can 
say “ a has the property P”. We know, for instance, that some one 
was Mr. Jones’s father, but we may be unable to say who he was. 
No one knows who was Napoleon Ill’s father, but we all believe 
that some one was. If a bullet whizzes past you when no one is 
in sight, you say “some one fired at me”. As a rule, in such cases, 
you are making an inference from a general proposition. Everyone 
has a father, therefore Mr. Jones has a father. If you believe that 
everything has a cause, many things will be only known to you 
as “the something that caused this”. Whether such generalizations 
are the only source of some-sentences not directly derived from 
perception, or whether, on the contrary, there must be some- 
sentences among the premisses of our knowledge, is a question 
which, for the present, I will leave open. 

There is a school, of which Brouwer is the founder, which 
holds that a some-sentence may be neither true nor false. The 
stock example is “there are three consecutive 7’s in the decimal 
expression of So far as this has been worked out, no three 
consecutive 7’s have occurred. If they occur at a later point, 
this may in time be discovered; but if they never occur, 
this can never be discovered. I have discussed this question 
in the “Inquiry into Meaning and Truth”, where I came to the 
conclusion that such sentences are always either true or false if 
they are syntactically significant. As I see no reason to change 
this view, I shall refer the reader to that book for a statement of 
my grounds, and I shall assume, without further argument, that 
all syntactically correct sentences are either true or false. 

Chapter XI 


T he purpose of this chapter is to state in dogmatic form 
certain conclusions which follow from previous discussions, 
together with the fuller discussions of “An Inquiry into 
Meaning and Truth”. More particularly, I wish to give meanings, 
as definite as possible, to the four words in the title of this chapter. 
I do not mean to deny that the words are susceptible of other 
equally legitimate meanings, but only that the meanings which I 
shall assign to them represent important concepts, which, when 
understood and distinguished, are useful in many philosophical 
problems, but when confused are a source of inextricable tangles. 


“Fact”, as I intend the term, can only be defined ostensively. 
Everything that there is in the world I call a “fact”. The sun is a 
fact ; Caesar’s crossing of the Rubicon was a fact ; if I have tooth- 
ache, my toothache is a fact. If I make a statement, my making it 
is a fact, and if it is true there is a further fact in virtue of which 
it is true, but not if it is false. The butcher says: “I’m sold out, 
and that’s a fact”; immediately afterwards, a favoured customer 
arrives, and gets a nice piece of lamb from under the counter. So 
the butcher told two lies, one in saying he was sold out, and the 
other in saying that his being sold out was a fact. Facts are what 
make statements true or false. I should like to confine the word 
“fact” to the minimum of what must be known in order that the 
truth or falsehood of any statement may follow analytically from 
those asserting that minimum. For example, if “Brutus was a 
Roman” and “Cassius was a Roman” each assert a fact, I should 
not say that “Brutus and Cassius were Romans” asserted a new 
fact. We have seen that the questions whether there are negative 
facts and general facts raise difficulties. These niceties, however, 
are largely linguistic. 

I mean by a “fact” something which is there, whether anybody 
thinks so or not. If I look up a railway time-table and find that 
there is a train to Edinburgh at io a.m., then, if the time-table is 


human knowledge: its scope and limits 

correct, there is an actual train, which is a “fact”. The statement 
in the time-table is itself a “fact”, whether true or false, but it 
only states a fact if it is true, i.e. if there really is a train. Most 
facts are independent of our volitions; that is why they are called 
“hard”, “stubborn”, or “ineluctable”. Physical facts, for the 
most part, are independent, not only of our volitions, but even 
of our existence. 

The whole of our cognitive life is, biologically considered, part 
of the process of adaptation to facts. This process is one which 
exists, in a greater or less degree, in all forms of life, but is not 
commonly called “cognitive” until it reaches a certain level of 
development. Since there is no sharp frontier anywhere between 
the lowest animal and the most profound philosopher, it is evident 
that we cannot say precisely at what point we pass from mere 
animal behaviour to something deserving to be dignified by the 
name of “ knowledge”. But at every stage there is adaptation, and 
that to which the animal adapts itself is the environment of fact . 


“Belief”, which we have next to consider, has an inherent and 
inevitable vagueness, which is due to the continuity of mental 
development from the amoeba to homo sapiens . In its most de- 
veloped form, which is that most considered by philosophers, it is 
displayed by the assertion of a sentence. After sniffing for a time, 
you exclaim: “Good heavens! the house is on fire.” Or, when a 
picnic is in contemplation, you say: “Look at those clouds: there 
will be rain.” Or, in a train, you try to subdue an optimistic fellow- 
passenger by observing: “Last time I did this journey we were 
three hours late.” Such remarks, if you are not lying, express 
beliefs. We are so accustomed to the use of words for expressing 
beliefs that it may seem strange to speak of “belief” in cases 
where there are no words. But it is clear that even when words 
are used they are not of the essence of the matter. The smell of 
burning first makes you believe that the house is on fire, and then 
the words come, not as being the belief, but as a way of putting 
it into a form of behaviour in which it can be communicated to 
others. I am thinking, of course, of beliefs that are not very com- 
plicated or refined. I believe that the angles of a polygon add up 
to twice as many right angles as the figure has sides diminished by 



four right angles, but a man would need super-human mathemati- 
cal intuition to be able to believe this without words. But the 
simpler kind of belief, especially when it calls for action, may be 
entirely unverbalized. When you are travelling with a companion, 
you may say: “We must run; the train is just going to start.” But 
if you are alone you may have the same belief, and run just as 
fast, without any words passing through your head. 

I propose, therefore, to treat belief as something that can be 
pre-intellectual, and can be displayed in the behaviour of animals. 
I incline to think that, on occasion, a purely bodily state may 
deserve to be called a “belief”. For example, if you walk into 
your room in the dark, and someone has put a chair in an unusual 
place, you may bump into it, because your body believed there 
was no chair there. But the parts played by mind and body 
respectively in belief are not very important to separate for our 
present purposes. A belief, as I understand the term, is a certain 
kind of state of body or mind or both. To avoid verbiage, I shall 
call it a state of an organism, and ignore the distinction of bodily 
and mental factors. 

One characteristic of a belief is that it has external reference, 
in the sense defined in a previous chapter. The simplest case, 
which can be observed behaviouristically, is when, owing to a 
conditioned reflex, the presence of A causes behaviour appropriate 
to B. This covers the important case of acting on information 
received: here the phrase heard is A, and what it signifies is B. 
Somebody says “look out, there’s a car coming”, and you act as 
you would if you saw the car. In this case you are believing what 
is signified by the phrase “a car is coming”. 

Any state of an organism which consists in believing something 
can, theoretically, be fully described without mentioning the 
something. When you believe “a car is coming”, your belief 
consists in a certain state of the muscles, sense-organs, and emo- 
tions, together perhaps with certain visual images. All this, and 
whatever else may go to make up your belief, could, in theory, 
be fully described by a psychologist and physiologist working 
together, without their ever having to mention anything outside 
your mind and body. Your state, when you believe that a car is 
coming, will be very different in different circumstances. You 
may be watching a race, and wondering whether the car on which 
you have put your money will win. You may be waiting for the 



HUMAN knowledge: its scope and limits 

return of your son from captivity in the Far East. You may be 
trying to escape from the police. You may be suddenly roused 
from absent-mindedness while crossing the street. But although 
your total state will not be the same in these various cases, there 
will be something in common among them, and it is this something 
which makes them all instances of the belief that a car is coming. 
A belief, we may say, is a collection of states of an organism 
bound together by all having, in whole or part, the same external 

In an animal or a young child, believing is shown by an action 
or series of actions. The beliefs of the hound about the fox are 
shown by his following the scent. But in human beings, as a 
result of language and of the practice of suspended reactions, 
believing often becomes a more or less static condition, consisting 
perhaps in pronouncing or imagining appropriate words, together 
with one of the feelings that constitute different kinds of belief. 
As to these, we may enumerate: first, the kind of belief that con- 
sists in filling out sensations by animal inferences; second, 
memory; third, expectation; fourth, the kind of belief generated 
unreflectingly by testimony; and fifth, the kind of belief resulting 
from conscious inference. Perhaps this list is both incomplete 
and in part redundant, but certainly perception, memory, and 
expectation differ as to the kinds of feeling involved. “Belief”, 
therefore, is a wide generic term, and a state of believing is not 
sharply separated from cognate states which would not naturally 
be described as believings. 

The question what it is that is believed when an organism is in 
a state of believing is usually somewhat vague. The hound pur- 
suing a scent is unusually definite, because his purpose is simple 
and he has no doubt as to the means; but a pigeon hesitating 
whether to eat out of your hand is in a much more vague and 
complex condition. Where human beings are concerned, language 
gives an illusory appearance of precision; a man may be able to 
express his belief in a sentence, and it is then supposed that the 
sentence is what he believes. But as a rule this is not the case. If 
you say “look, there is Jones”, you are believing something, and 
expressing your belief in words, but what you are believing has 
to do with Jones, not with the name “Jones”. You may, on another 
occasion, have a belief which is concerned with words. “Who is 
that very distinguished man who has just come in ? That is Sir 



Theophilus Thwackum.” In this case it is the name you want. 
But as a rule in ordinary speech the words are, so to speak, trans- 
parent; they are not what is believed, any more than a man is the 
name by which he is called. 

When words merely express a belief which is about what the 
words mean, the belief indicated by the words is lacking in 
precision to the degree that the meaning of the words is lacking 
in precision. Outside logic and pure mathematics, there are no 
words of which the meaning is precise, not even such words as 
“centimetre” and “second”. Therefore even when a belief is 
expressed in words having the greatest degree of precision of 
which empirical words are capable, the question as to what it is 
that is believed is still more or less vague. 

This vagueness does not cease when a belief is what may be 
called “purely verbal”, i.e. when what is believed is that a certain 
sentence is true. This is the sort of belief acquired by schoolboys 
whose education has been on old-fashioned lines. Consider the 
difference in the schoolboy s attitude to “William the Conqueror, 
1066” and “next Wednesday will be a whole holiday”. In the 
former case, he knows that that is the right form of words, and 
cares not a pin for their meaning; in the latter case, he acquires 
a belief about next Wednesday, and cares not a pin what words 
you use to generate his belief. The former belief, but not the 
latter, is “purely verbal”. 

If I were to say that the schoolboy is believing that the sentence 
“William the Conqueror, 1066” is “true”, I should have to add 
that his definition of “truth” is purely pragmatic: a sentence is 
“true” if the consequences of uttering it in the presence of a 
master are pleasant; if they are unpleasant, it is “false”. 

Forgetting the schoolboy, and resuming our proper character 
as philosophers, what do we mean when we say that a certain 
sentence is “true”? I am not yet asking what is meant by “true”; 
this will be our next topic. For the moment I am concerned to 
point out that, however “true” may be defined, the significance 
of “this sentence is true” must depend upon the significance of 
the sentence, and is therefore vague in exactly the degree in which 
there is vagueness in the sentence which is said to be true. We do 
not therefore escape from vagueness by concentrating attention 
on purely verbal beliefs. 

Philosophy, like science, should realize that, while complete 


HUMAN knowledge: its scope and limits 

precision is impossible, techniques can be invented which 
gradually diminish the area of vagueness or uncertainty. However 
admirable our measuring apparatus may be, there will always 
remain some lengths concerning which we are in doubt whether 
they are greater than, less than, or equal to, a metre; but there 
is no known limit to the refinements by which the number of such 
doubtful lengths can be diminished. Similarly, when a belief is 
expressed in words, there will always remain a band of possible 
circumstances concerning which we cannot say whether they 
would make the belief true or false, but the breadth of this band 
can be indefinitely diminished, partly by improved verbal analysis, 
partly by a more delicate technique in observation. Whether 
complete precision is or is not theoretically possible depends 
upon whether the physical world is discrete or continuous. 

Let us now consider the case of a belief expressed in words all 
of which have the greatest attainable degree of precision. Suppose, 
for the sake of concreteness, that I believe the sentence: “My 
height is greater than 5 ft. 8 ins. and less than 5 ft. 9 ins.” Let 
us call this sentence “S”. I am not yet asking what would make 
this sentence true, or what would entitle me to say that I know it; 
I am asking only: “What is happening in me when I have the 
belief which I express by the sentence S?” There is obviously no 
one correct answer to this question. All that can be said definitely 
is that I am in a state such as, if certain further things happen, 
will give me a feeling which might be expressed by the words 
“quite so”, and that, now, while these things have not yet happened, 
I have the idea of their happening combined with the feeling 
expressed by the word “yes”. I may, for instance, imagine myself 
standing against a wall on which there is a scale of feet and inches, 
and in imagination see the top of my head between two marks on 
this scale, and towards this image I may have the feeling of assent. 
We may take this as the essence of what may be called “static” 
belief, as opposed to belief shown by action : static belief consists 
in an idea or image combined with a yes-feeling. 


I come now to the definition of “truth” and “falsehood”. 
Certain things are evident. Truth is a property of beliefs, and 
derivatively of sentences which express beliefs. Truth consists 



in a certain relation between a belief and one or more facts other 
than the belief. When this relation is absent, the belief is false. 
A sentence may be called “true” or “false” even if no one believes 
it, provided that, if it were believed, the belief would be true or 
false as the case may be. 

So much, I say, is evident. But what is not evident is the nature 
of the relation between belief and fact that is involved, or the 
definition of the possible fact that will make a given belief true, or 
the meaning of “possible” in this phrase. Until these questions 
are answered we have no adequate definition of “truth”. 

Let us begin with the biologically earliest form of belief, which 
is to be seen among animals as among men. The compresence of 
two kinds of circumstance, A and B, if it has been frequent or 
emotionally interesting, is apt to have the result that, when A is 
sensibly present, the animal reacts as it formerly reacted to B, or 
at any rate displays some part of this reaction. In some animals 
this connection may be sometimes innate, and not the result of 
experience. But however the connection may be brought about, 
when the sensible presence of A causes acts appropriate to B, we 
may say that the animal “believes” B to be in the environment, 
and that the belief is “true” if B is in the environment. If you wake 
a man up in the middle of the night and shout “fire!” he will 
leap from his bed even if he does not yet see or smell fire. His 
action is evidence of a belief which is “true” if there is fire, and 
“false” otherwise. Whether his belief is true depends upon a fact 
which may remain outside his experience. He may escape so fast 
that he never acquires sensible evidence of the fire; he may fear 
that he will be suspected of incendiarism and flee the country, with- 
out ever inquiring whether there was a fire or not; nevertheless 
his belief remains true if there was the fact (namely fire) which 
constituted its external reference or significance, and if there was 
not such a fact his belief remained false even if all his friends 
assured him that there had been a fire. 

The difference between a true and false belief is like that be- 
tween a wife and a spinster : in the case of a true belief there is a 
fact to which it has a certain relation, but in the case of a false 
belief there is no such fact. To complete our definition of “truth” 
and “falsehood” we need a description of the fact which would 
make a given belief true, this description being one which applies 
to nothing if the belief is false. Given a woman of whom we do not 


HUMAN knowledge: its scope and limits 

know whether she is married or not, we can frame a description 
which will apply to her husband if she has one, and to nothing 
if she is a spinster. Such a description would be: “the man who 
stood beside her in a church or registry office while certain words 
were pronounced”. In like manner we want a description of the 
fact or facts which, if they exist, make a belief true. Such fact or 
facts I call the “verifier” of the belief. 

What is fundamental in this problem is the relation between 
sensations and images, or, in Hume’s terminology, between 
impressions and ideas. We have considered in a previous chapter 
the relation of an idea to its prototype, and have seen how 
“meaning” develops out of this relation. But given meaning and 
syntax, we arrive at a new concept, which I call “significance”, 
and which is characteristic of sentences and of complex images. 
In the case of single words used in an exclamatory manner, such 
as “fire!” or “murder!” meaning and significance coalesce, but 
in general they are distinct. The distinction is made evident by 
the fact that words must have meaning if they are to serve a 
purpose, but a string of words does not necessarily have signifi- 
cance. Significance is a characteristic of all sentences that are not 
nonsensical, and not only of sentences in the indicative, but also 
of such as are interrogative, imperative, or optative. For present 
purposes, however, we may confine ourselves to sentences in the 
indicative. Of these we may say that the significance consists in 
the description of the fact which, if it exists, will make the sentence 
true. It remains to define this description. 

Let us take an illustration. Jefferson had a belief expressed in 
the words: “There are mammoths in North America.” This 
belief might have been true even if no one had seen one of these 
mammoths ; there might, when he expressed the belief, have been 
just two in an uninhabited part of the Rocky Mountains, and they 
might soon afterwards have been swept by a flood down the 
Colorado River into the sea. In that case, in spite of the truth of his 
belief, there would have been no evidence for it. The actual 
mammoths would have been facts, and would have been, in the 
above sense, “verifiers” of the belief. A verifier which is not 
experienced can often be described, if it has a relation known by 
experience to something known by experience; it is in this way 
that we understand such a phrase as “the father of Adam”, which 
describes nothing. It is in this way that we understand Jefferson’s 

1 66 


belief about mammoths: we know the sort of facts that would 
have made his belief true, that is to say, we can be in a state of 
mind such that, if we had seen mammoths, we should have 
exclaimed: “Yes, that’s what I was thinking of.” 

The significance of a sentence results from the meanings of its 
words together with the laws of syntax. Although meanings must 
be derived from experience, significance need not. I know from 
experience the meaning of “man” and the meaning of “wings”, 
and therefore the significance of the sentence “There is a winged 
man”, although I have no experience of what this sentence signi- 
fies. The significance of a sentence may always be understood as 
in some sense a description. When this description describes a 
fact, the sentence is “true”; otherwise it is “false”. 

It is important not to exaggerate the part played by convention. 
So long as we are considering beliefs, not the sentences in which 
they are expressed, convention plays no part at all. Suppose you 
are expecting to meet some person of whom you are fond, and 
whom you have not seen for some time. Your expectation may 
be quite wordless, even if it is detailed and complex. You may 
hope that he will be smiling, you may recall his voice, his gait, the 
expression of his eyes ; your total expectation may be such as only 
a good painter could express, in paint, not in words. In this case 
you are expecting an experience of your own, and the truth or 
falsehood of your expectation is covered by the relation of idea 
and impression : your expectation is “true” if the impression, when 
it comes, is such that it might have been the prototype of your 
previous idea if the time-order had been reversed. This is what 
we express when we say: “That is what I expected to see.” Con- 
vention is concerned only in the translation of belief into language, 
or (if we are told something) of language into belief. Moreover 
the correspondence of language and belief, except in abstract 
matters, is usually by no means exact: the belief is richer in detail 
and context than the sentence, which picks out only certain 
salient features. You say “I shall see him soon”, but you think “I 
shall see him smiling, but looking older, friendly, but shy, with 
his hair untidy and his shoes muddy” — and so on, through 
an endless variety of detail of which you may be only half 

The case of an expectation is the simplest from the point of 
view of defining truth and falsehood, for in this case the fact 


HUMAN knowledge: its scope and limits 

upon which truth or falsehood depends is about to be experienced. 
Other cases are more difficult. 

Memory, from the standpoint of our present problem, is closely 
analogous to expectation. A recollection is an idea, while the fact 
recollected was an impression; the memory is “true” if the re- 
collection has to the fact that kind of resemblance which exists 
between an idea and its prototype. 

Consider, next, such a statement as “you have a toothache”. 
In any belief concerning another person’s experience there may 
be the same sort of extra-verbal richness that we have seen to 
be frequent in regard to expectations of our own experiences; 
you may, having recently had toothache, feel sympathetically the 
throbbing pangs that you imagine your friend to be suffering. 
Whatever wealth or paucity of imagination you may bring to bear, 
it is clear that your belief is “true” in proportion as it resembles 
the fact of your friend’s toothache — the resemblance being again 
of the sort that can subsist between idea and prototype. 

But when we pass on to something which no one experiences 
or has experienced, such as the interior of the earth, or the world 
before life began, both belief and truth become more abstract 
than in the above cases. We must now consider what can be 
meant by “truth” when the verifying fact is experienced by no 

Anticipating coming discussions, I shall assume that the 
physical world, as it is independently of perception, can be known 
to have a certain structural similarity to the world of our percepts, 
but cannot be known to have any qualitative similarity. And when 
I say that it has structural similarity, I am assuming that the 
ordering relations in terms of which the structure is defined are 
spatio-temporal relations such as we know in our own experience. 
Certain facts about the physical world, therefore — those facts, 
namely, which consist of space-time structure — are such as we can 
imagine. On the other hand, facts as to the qualitative character 
of physical occurences are, presumably, such as we cannot 

Now while there is no difficulty in supposing that there are 
unimaginable facts, there cannot be beliefs , other than general 
beliefs, of which the verifiers would be unimaginable. This is an 
important principle, but if it is not to lead us astray a little care is 
necessary as regards certain logical points. The first of these is 

1 68 


that we may know a general proposition although we do not know 
any instance of it. On a large pebbly beach you may say, probably 
with truth: “There are pebbles on this beach which no one will 
ever have noticed.” It is quite certainly true that there are finite 
integers which no one will ever have thought of. But it is self- 
contradictory to suppose such propositions established by giving 
instances of their truth. This is only an application of the principle 
that we can understand statements about all or some of the 
members of a class without being able to enumerate the members. 
We understand the statement “all men are mortal” just as com- 
pletely as we should if we could give a complete list of men; for 
to understand this statement we need only understand the concepts 
“man” and “mortal” and what is meant by being an instance of 

Now take the statement: “There are facts which I cannot 
imagine.” I am not considering whether this statement is true; 
I am only concerned to show that it is intelligible. Observe, in 
the first place, that if it is not intelligible, its contradictory must 
also be not intelligible, and therefore not true, though also not 
false. Observe, in the second place, that to understand the state- 
ment it is unnecessary to be able to give instances, any more than 
of the unnoticed pebbles or the numbers that are not thought of. 
All that is necessary is to understand the words and the syntax, 
which we do. The statement is therefore intelligible; whether it 
is true is another matter. 

Take, now, the following statement: “There are electrons, but 
they cannot be perceived.” Again I am not asking whether the 
statement is true, but what is meant by supposing it true or be- 
lieving it to be true. “Electron” is a term defined by means of 
causal and spatio-temporal relations to events that we experience, 
and to other events related to them in ways of which we have 
experience. We have experience of the relation “parent”, and can 
therefore understand the relation “great-great-great-grandparent”, 
although we have no experience of this relation. In like manner 
we can understand sentences containing the word “electron”, in 
spite of not perceiving anything to which this word is applicable. 
And when I say we can understand such sentences, I mean that 
we can imagine facts which would make them true. 

The peculiarity, in such cases, is that we can imagine general 
circumstances which would verify our belief, but cannot imagine 


HUMAN knowledge: its scope and limits 

the particular facts which are instances of the general fact. I 
cannot imagine any particular fact of the form: “w is a number 
which will never have been thought of”, for, whatever value 1 
give to w, my statement becomes false by the very fact of my 
giving that value. But I can quite well imagine the general fact 
which gives truth to the statement: “There are numbers which 
will never have been thought of.” The reason is that general 
statements are concerned with intensions, and can be understood 
without any knowledge of the corresponding extensions. 

Beliefs as to what is not experienced, as the above discussion 
has shown, are not as to unexperienced individuals, but as to 
classes of which no member is experienced. A belief must always 
be capable of being analysed into elements that experience has 
made intelligible, but when a belief is set out in logical form it 
often suggests a different analysis, which would seem to involve 
components not known by experience. When such psychologically 
misleading analysis is avoided, we can say, quite generally: Every 
belief which is not merely an impulse to action is in the nature of 
a picture, combined with a yes-feeling or a no-feeling; in the case 
of a yes-feeling it is “true” if there is a fact having to the picture 
the kind of similarity that a prototype has to an image; in the 
case of a no-feeling it is “true” if there is no such fact. A belief 
which is not true is called “false”. 

This is a definition of “truth” and “falsehood”. 


I come now to the definition of “knowledge”. As in the cases 
of “belief” and “truth”, there is a certain inevitable vagueness and 
inexactitude in the conception. Failure to realize this has led, it 
seems to me, to important errors in the theory of knowledge. 
Nevertheless, it is well to be as precise as possible about the un- 
avoidable lack of precision in the definition of which we are in 

It is clear that knowledge is a sub-class of true beliefs: every 
case of knowledge is a case of true belief, but not vice versa. It 
is very easy to give examples of true beliefs that are not knowledge. 
There is the man who looks at a clock which is not going, though 
he thinks it is, and who happens to look at it at the moment when 
it is right; this man acquires a true belief as to the time of day, 



but cannot be said to have knowledge. There is the man who 
believes, truly, that the last name of the Prime Minister in 1906 
began with a B, but who believes this because he thinks that 
Balfour was Prime Minister then, whereas in fact it was Campbell- 
Bannerman. There is the lucky optimist who, having bought a 
ticket for a lottery, has an unshakeable conviction that he will win, 
and, being lucky, does win. Such instances can be multiplied 
indefinitely, and show that you cannot claim to have known 
merely because you turned out to be right. 

What character in addition to truth must a belief have in order 
to count as knowledge? The plain man would say there must be 
sound evidence to support the belief. As a matter of common 
sense this is right in most of the cases in which doubt arises in 
practice, but if intended as a complete account of the matter it is 
very inadequate. “Evidence” consists, on the one hand, of certain 
matters of fact that are accepted as indubitable, and, on the other 
hand, of certain principles by means of which inferences are 
drawn from the matters of fact. It is obvious that this process is 
unsatisfactory unless we know the matters of fact and the principles 
of inference not merely by means of evidence, for otherwise we 
become involved in a vicious circle or an endless regress. We 
must therefore concentrate our attention on the matters of fact 
and the principles of inference. We may then say that what is 
known consists, first, of certain matters of fact and certain prin- 
ciples of inference, neither of which stands in need of extraneous 
evidence, and secondly, of all that can be ascertained by applying 
the principles of inference to the matters of fact. Traditionally, 
the matters of fact are those given in perception and memory, 
while the principles of inference are those of deductive and in- 
ductive logic. 

There are various unsatisfactory features in this traditional 
doctrine, though I am not at all sure that, in the end, we can 
substitute anything very much better. In the first place, the 
doctrine does not give an intensional definition of “knowledge”, 
or at any rate not a purely intensional definition; it is not clear 
what there is in common between facts of perception and prin- 
ciples of inference. In the second place, as we shall see in Part III, 
it is very difficult to say what are facts of perception. In the third 
place, deduction has turned out to be much less powerful than was 
formerly supposed ; it does not give new knowledge, except as to 


human knowledge: its scope and limits 

new forms of words for stating truths in some sense already 
known. In the fourth place, the methods of inference that may 
be called in a broad sense “inductive” have never been satis- 
factorily formulated; when formulated, even if completely true, 
they only give probability to their conclusions; moreover, in any 
possibly accurate form, they lack self-evidence, and are only to be 
believed, if at all, because they seem indispensable in reaching 
conclusions that we all accept. 

There are, broadly speaking, three ways that have been sugges- 
ted for coping with the difficulties in defining “knowledge”. The 
first, and oldest, is to emphasize the concept of “self-evidence”. 
The second is to abolish the distinction between premisses and 
conclusions, and to say that knowledge is constituted by the 
coherence of a whole body of beliefs. The third and most drastic 
is to abandon the concept of “knowledge” altogether and substi- 
tute “beliefs that promote success” — and here “success” may 
perhaps be interpreted biologically. We may take Descartes, 
Hegel, and Dewey as protagonists of these three points of view. 

Descartes holds that whatever I conceive clearly and distinctly 
is true. He believes that, from this principle, he can derive not 
only logic and metaphysics, but also matters of fact, at least in 
theory. Empiricism has made such a view impossible; we do not 
think that even the utmost clarity in our thoughts would enable 
us to demonstrate the existence of Cape Horn. But this does not 
dispose of the concept of “self-evidence”: we may say that what 
he says applies to conceptual evidence, but that there is also 
perceptual evidence, by means of which we come to know matters 
of fact. I do not think we can entirely dispense with self-evidence. 
If you slip on a piece of orange peel and hit your head with a 
bump on the pavement, you will have little sympathy with a 
philosopher who tries to persuade you that it is uncertain whether 
you are hurt. Self-evidence also makes you accept the argument 
that if all men are mortal and Socrates is a man, then Socrates is 
mortal. I do not know whether self-evidence is anything except a 
certain firmness of conviction ; the essence of it is that, where it is 
present, we cannot help believing. If, however, self-evidence is to 
be accepted as a guarantee of truth, the concept must be carefully 
distinguished from others that have a subjective resemblance to 
it. I think we must bear it in mind as relevant to the definition of 
“knowledge”, but as not in itself sufficient. 



Another difficulty about self-evidence is that it is a matter of 
degree. A clap of thunder is indubitable, but a very faint noise is 
not; that you are seeing the sun on a bright day is self-evident, 
but a vague blur in a fog may be imaginary ; a syllogism in Barbara 
is obvious, but a difficult step in a mathematical argument may 
be very hard to “see”. It is only for the highest degree of self- 
evidence that we should claim the highest degree of certainty. 

The coherence theory and the instrumentalist theory are 
habitually set forth by their advocates as theories of truth . As 
such they are open to certain objections which I have urged else- 
where. I am considering them now, not as theories of truths but 
as theories of knowledge. In this form there is more to be said for 

Let us ignore Hegel, and set forth the coherence theory of 
knowledge for ourselves. We shall have to say that sometimes two 
beliefs cannot both be true, or, at least, that we sometimes believe 
this. If I believe simultaneously that A is true, that B is true, and 
that A and B cannot both be true, I have three beliefs which do 
not form a coherent group. In that case at least one of the three 
must be mistaken. The coherence theory in its extreme form 
maintains that there is only one possible group of mutually co- 
herent beliefs, which constitutes the whole of knowledge and the 
whole of truth. I do not believe this; I hold, rather, to Leibniz’s 
multiplicity of possible worlds. But in a modified form the co- 
herence theory can be accepted. In this modified form it will say 
that all, or nearly all, of what passes for knowledge is in a greater 
or less degree uncertain ; that, if principles of inference are among 
the prima facie materials of knowledge, then one piece of prima 
facie knowledge may be inferrible from another, and thus acquires 
more credibility than it had on its own account. It may thus 
happen that a body of propositions, each of which has only a 
moderate degree of credibility on its own account, may collectively 
have a very high degree of credibility. But this argument depends 
upon the possibility of varying degrees of intrinsic credibility, and 
is therefore not a pure coherence theory. I shall consider this 
matter in more detail in Part V. 

With respect to the theory that we should substitute for “know- 
ledge” the concept “beliefs that promote success”, it is sufficient 
to point out that it derives whatever plausibility it may possess 
from being half-hearted. It assumes that we can know (in the old- 


HUMAN knowledge: its scope and limits 

fashioned sense) what beliefs promote success, for if we cannot 
know this the theory is useless in practice, whereas its purpose 
is to glorify practice at the expense of theory. In practice, ob- 
viously, it is often very difficult to know what beliefs promote 
success, even if we have an adequate definition of “success”. 

The conclusion to which we seem to be driven is that knowledge 
is a matter of degree. The highest degree is found in facts of 
perception, and in the cogency of very simple arguments. The 
next highest degree is in vivid memories. When a number of 
beliefs are each severally in some degree credible, they become 
more so if they are found to cohere as a logical whole. General 
principles of inference, whether deductive or inductive, are 
usually less obvious than many of their instances, and are psycho- 
logically derivative from apprehension of their instances. Towards 
the end of our inquiry I shall return to the definition of “know- 
ledge”, and shall then attempt to give more precision and articula- 
tion to the above suggestions. Meanwhile let us remember that 
the question “what do we mean by ‘knowledge’?” is not one to 
which there is a definite and unambiguous answer, any more than 
to the question “what do we mean by ‘baldness’?” 





W E come now to an inquiry which proceeds in the opposite 
order from that of our initial survey of the universe. In 
that survey we were attempting to be as far as possible 
impartial and impersonal; it was our aim to come as near as our 
capacities permit to describing the world as it might appear to an 
observer of miraculous perceptive powers viewing it from without. 
We were concerned with what we know rather than with what we 
know. We attempted to use an order in our description which 
ignored, for the moment, the fact that we are part of the universe, 
and that any account which we can give of it depends upon its 
effects upon ourselves, and is to this extent inevitably anthropo- 
centric. We accordingly began with the system of galaxies, and 
passed on, by stages, to our own galaxy, our own little solar 
system, our own tiny planet, the infinitesimal specks of life upon 
its surface, and finally, as the climax of insignificance, the bodies 
and minds of those odd beings that have imagined themselves the 
lords of creation and the end and aim of the whole vast cosmos. 

But this survey, which seems to end in the pettiness of Man and 
all his concerns, is only one side of the truth. There is another side, 
which must be brought out by a survey of a different kind. In this 
second kind of survey, which is now to occupy our attention, we 
ask no longer what the universe is, but how we come to know 
whatever we do know about it. In this survey Man again occupies 
the centre, as in the Ptolemaic astronomy. What we know of the 
world we know by means of events in our own lives, events which, 
but for the power of thought, would remain merely private. 
The little dots that an astronomer sees on a photographic plate are 
to him signs of vast galaxies separated from him by hundreds of 
thousands of light-years. All the immensities of space and all the 
abysses of time are mirrored in his thought, which, in a sense, 
is as vast as they are. Nothing is too great or too small for his 
intellect to comprehend, nothing is too distant in time or space 
for him to assign to it its due weight in the structure of the cosmos. 
In power he is nearly as feeble as his minuteness suggests, but in 
contemplation he is boundless, and the equal of all that he can 

It is my purpose in the following Parts to discuss, first our data, 

177 M 

HUMAN knowledge: its scope and limits 

and then the relation ofscience to the crude material of experience. 
The data from which scientific inferences proceed are private to 
ourselves; what we call “seeing the sun” is an event in the life of 
the seer, from which the astronomer’s sun has to be inferred by a 
long and elaborate process. It is evident that, if the world were a 
higgledy-piggledy chaos, inferences of this kind would be im- 
possible; but for causal interconnectedness, what happens in one 
place would afford no indication of what has happened in another, 
and my experiences would tell me nothing of events outside my 
own biography. It is the process from private sensation and 
thought to impersonal science that will now concern us. The road 
is long and rugged, and the goal must be kept in view if the 
journey is not to seem wearisome. But until we have traversed 
this road neither the scope nor the essential limitations of human 
knowledge can be adequately understood. 

The inferences upon which we implicitly rely in this investiga- 
tion, of which the explicit logic will be considered in Part VI, 
differ from those of deductive logic and mathematics in being not 
demonstrative, i.e. in being inferences which, when the premisses 
are true and the reasoning correct, do not insure the truth of the 
conclusion, though they arc held to make the conclusion “prob- 
able” in some sense and in some degree. Except in mathematics, 
almost all the inferences upon which we actually rely are of this 
sort. In some cases the inference is so strong as to amount to 
practical certainty. A page of typescript which makes sense is 
assumed to have been typed by someone, although, as Eddington 
points out, it may have been produced accidentally by a monkey 
walking on a typewriter, and this bare possibility makes the 
inference to an intentional typist non-demonstrative. Many 
inferences which are accepted by all men of science are much less 
nearly certain, for instance, the theory that sound is transmitted 
by waves. There is a gradation in the probability assigned to 
different inferences by scientific common sense, but there is no 
accepted body of principles according to which such probabilities 
are to be estimated. I should wish, by analysing scientific pro- 
cedure, to systematize the rules of such inference. The ideal 
would be the kind of systematization which has been achieved in 
relation to deductive logic. 

It has been customary to regard all inference as deductive or 
inductive, and to regard probable inference as synonymous with 

i 7 8 


inductive inference. I believe that, if ordinarily accepted scientific 
inferences are to be accepted as valid, we shall have need of other 
principles in addition to induction if not in place of it. 

We may take three questions as typical of those that I wish to 
investigate. These three are as to the best available grounds for 
believing: (i) that the world existed yesterday, ( 2 ) that the sun 
will rise to-morrow, (3) that there are sound-waves. I am not 
asking whether these beliefs are true, but what, assuming them 
true, are the best reasons for believing them. And generally: why 
should we believe things asserted by science but not verified by 
present perception? The answer, if I am not mistaken, is by no 
means simple. 


Chapter I 


W HEN we examine our beliefs as to matters of fact, we find 
that they are sometimes based directly on perception or 
memory, while in other cases they are inferred. To com- 
mon sense this distinction presents little difficulty: the beliefs that 
arise immediately from perception appear to it indubitable, and the 
inferences, though they may sometimes be wrong, are thought, in 
such cases, to be fairly easily rectified except where peculiarly 
dubious matters are concerned. I know of the existence of my 
friend Mr. Jones because I see him frequently: in his presence 
I know him by perception, and in his absence by memory. I know 
of the existence of Napoleon because 1 have heard and read about 
him, and I have every reason to believe in the veracity of my 
teachers. I am somewhat less certain about Hengist and Horsa, 
and much less certain about Zoroaster, but these uncertainties 
are still on a common-sense level, and do not seem, at first sight, 
to raise any philosophical issue. 

This primitive confidence, however, was lost at a very early 
stage in philosophical speculation, and was lost for sound reasons. 
It was found that what I know by perception is less than has been 
thought, and that the inferences by which I pass from perceived 
to unperceived facts are open to question. Both these sources of 
scepticism must be investigated. 

There is, to begin with, a difficulty as to what is inferred and 
what is not. I spoke a moment ago of my belief in Napoleon as an 
inference from what I have heard and read, but there is an im- 
portant sense in which this is not quite true. When a child is 
being taught history, he does not argue: “My teacher is a person 
of the highest moral character, paid to teach me facts; my teacher 
says there was such a person as Napoleon; therefore probably 
there was such a person/’ If he did, he would retain considerable 
doubt, since his evidence of the teacher’s moral character is likely 
to be inadequate, and in many countries at many times teachers 
have been paid to teach the opposite of facts. The child in fact, 
unless he hates the teacher, spontaneously believes what he is told. 



When we are told anything emphatically or authoritatively, it is 
an effort not to believe it, as any one can experience on April 
Fools* day. Nevertheless there is still a distinction, even on a 
common-sense level, between what we are told and what we know 
for ourselves. If you say to the child “how do you know about 
Napoleon?” the child may say “because my teacher told me”. 
If you say “how do you know your teacher told you”? the child 
may say “why, of course, because I heard her**. If you say “how 
do you know you heard her”? he may say “because I remember 
it distinctly”. If you say “how do you know you remember it”? 
he will either lose his temper or say “well, I do remember it”. 
Until you reach this point, he will defend his belief as to a matter 
of fact by belief in another matter of fact, but in the end he reaches 
a belief for which he can give no further reason. 

There is thus a distinction between beliefs that arise spon- 
taneously and beliefs for which no further reason can be given. It 
is the latter class of beliefs that are of most importance for theory 
of knowledge, since they are the indispensable minimum of pre- 
misses for our knowledge of matters of fact. Such beliefs I shall 
call “data”. In ordinary thinking they are causes of other beliefs 
rather than premisses from which other beliefs are inferred ; but 
in a critical scrutiny of our beliefs as to matters of fact we must, 
wherever possible, translate the causal transitions of primitive 
thinking into logical transitions, and only accept the derived 
beliefs to the extent that the character of the transitions seems to 
justify. For this there is a common-sense reason, namely, that 
every such transition is found to involve some risk of error, and 
therefore data are more nearly certain than beliefs derived from 
them. I am not contending that data are ever completely certain, 
nor is this contention necessary for their importance in theory 
of knowledge. 

There is a long history of discussions as to what was mistakenly 
called “scepticism of the senses”. Many appearances are deceptive. 
Things seen in a mirror may be thought to be “rear*. In certain 
circumstances, people see double. The rainbow seems to touch 
the ground at some point, but if you go there you do not find it. 
Most noteworthy in this connection are dreams: however vivid 
they may have been, we believe, when we wake up, that the 
objects which we thought we saw were illusory. But in all these 
cases the core of data is not illusory, but only the derived beliefs. 


HUMAN knowledge: its scope and limits 

My visual sensations, when I look in a mirror or see double, are 
exactly what I think they are. Things at the foot of the rainbow 
do really look coloured. In dreams I have all the experiences that 
I seem to have; it is only things outside my mind that are not as I 
believe them to be while I am dreaming. There are in fact no 
illusions of the senses, but only mistakes in interpreting sensa- 
tional data as signs of things other than themselves. Or, to speak 
more exactly, there is no evidence that there are illusions of the 

Every sensation which is of a familiar kind brings with it 
various associated beliefs and expectations. When (say) we see 
and hear an aeroplane, we do not merely have a visual sensation 
and the auditory sensation of a whirring noise ; spontaneously and 
without conscious thought we interpret what we see and hear and 
fill it out with customary adjuncts. To what an extent we do this 
becomes obvious when we make a mistake, for example when what 
we thought was an aeroplane turns out to be a bird. I knew a road, 
along which I used often to go in a car, which had a bend at a 
certain place, and a white-washed wall straight ahead. At night 
it was very difficult not to see the wall as a road going straight on 
up a hill. The right interpretation as a house and the wrong inter- 
pretation as an up-hill road were both, in a sense, inferences from 
the sensational datum, but they were not inferences in the logical 
sense, since they occurred without any conscious mental process. 

I give the name “animal inference” to the process of spontaneous 
interpretation of sensations. When a dog hears himself called in 
tones to which he is accustomed, he looks round and runs in the 
direction of the sound. He may be deceived, like the dog looking 
into the gramophone in the advertisement of “His Master's 
Voice”. But since inferences of this sort are generated by the 
repeated experiences that give rise to habit, his inference must 
be one which has usually been right in his past life, since other- 
wise the habit would not have been generated. We thus find 
ourselves, when we begin to reflect, expecting all sorts of things 
that in fact happen, although it would be logically possible for 
them not to happen in spite of the occurrence of the sensations 
which give rise to the expectations. Thus reflection upon animal 
inference gives us an initial store of scientific laws, such as “dogs 
bark”. These initial laws are usually somewhat unreliable, but 
they help us to take the first steps towards science. 



Every-day generalizations, such as “dogs bark”, come to be 
explicitly believed after habits have been generated which might 
be described as a pre-verbal form of the same belief. What sort 
of habit is it that comes to be expressed in the words “dogs bark” ? 
We do not expect them to bark at all times, but we do expect that, 
if they make a noise, it will be a bark or a growl. Psychologically, 
induction does not proceed as it does in the text-books, where we 
are supposed to have observed a number of occasions on which 
dogs barked, and then proceeded consciously to generalize. The 
fact is that the generalization, in the form of a habit of expectation, 
occurs at a lower level than that of conscious thought, so that, 
when we begin to think consciously, we find ourselves believing 
the generalization, not, explicitly, on the basis of the evidence, 
but as expressing what is implicit in our habit of expectation. 
This is a history of the belief, not a justification of it. 

Let us make this state of affairs somewhat more explicit. First 
comes the repeated experience of dogs barking, then comes the 
habit of expecting a bark, then, by giving verbal expression to 
the habit, comes belief in the general proposition “dogs bark”. 
Last comes the logician, who asks, not “why do I believe this”? 
but “what reason is there for supposing this true”? Clearly the 
reason, if any, must consist of two parts: first, the facts of per- 
ception consisting of the various occasions on which we have 
heard dogs bark; second, some principle justifying generalization 
from observed instances to a law. But this logical process comes 
historically after, not before, our belief in a host of common-sense 

The translation of animal inferences into verbal generalizations 
is carried out very inadequately in ordinary thinking, and even in 
the thinking of many philosophers. In what counts as perception 
of external objects there is much that consists of habits generated 
by past experience. Take, for example, our belief in the perma- 
nence of objects. When we see a dog or a cat, a chair or a table, we 
do not suppose that we are seeing something which has a merely 
momentary existence; we are convinced that what we are seeing 
has a past and a future of considerable duration. We do not think 
this about everything that we see ; a flash of lightning, a rocket, 
or a rainbow is expected to disappear quickly. But experience has 
generated in us the expectation that ordinary solid objects, which 
can be touched as well as seen, usually persist, and can be seen 


HUMAN knowledge: its scope and limits 

and touched again on suitable occasions. Science reinforces this 
belief by explaining away apparent disappearances as transforma- 
tions into gaseous forms. But the belief in quasi-permanence, 
except in exceptional cases, antedates the scientific doctrine of the 
indestructibility of matter, and is itself antedated by the animal 
expectation that common objects can be seen again if we look 
in the right place. 

The filling out of the sensational core by means of animal 
inferences, until it becomes what is called “ perception”, is ana- 
logous to the filling out of telegraphic press messages in news- 
paper offices. The reporter telegraphs the one word “King”, and 
the newspaper prints “His Gracious Majesty King George VI”. 
There is some risk of error in this proceeding, since the reporter 
may have been relating the doings of Mr. Mackenzie King. It is 
true that the context would usually reveal such an error, but one 
can imagine circumstances in which it would not. In dreams, we 
fill out the bare sensational message wrongly, and only the context 
of waking life shows us our mistake. 

The analogy to abbreviated press telegrams is very close. 
Suppose, for instance, you see a friend at the window of an in- 
coming train, and a little later you see him coming towards you 
on the platform. The physical causes of your perceptions (and 
of your interpretation of them) are certain light-signals passing 
between him and your eyes. All that physics, by itself, entitles 
you to infer from the receipt of these signals is that, somewhere 
along the line ol sight, light of the appropriate colours has been 
emitted or reflected or refracted or scattered. It is obvious that 
the kind of ingenuity which has produced the cinema could cause 
you to have just these sensations in the absence of your friend, and 
that in that case you would be deceived. But such sources of 
deception cannot be frequent, or at least cannot have been frequent 
hitherto, since, if they were, you would not have formed the 
habits of expectation and belief in context that you have in fact 
formed. In the case supposed, you are confident that it is your 
friend, that he has existed throughout the interval between seeing 
him at the window and seeing him on the platform, and that he has 
pursued a continuous path through space from the one to the 
other. You have no doubt that what you saw was something solid, 
not an intangible object like a rainbow or a cloud. And so, although 
the message received by the senses contains (so to speak) only a 



few key words, your mental and physical habits cause you, spon- 
taneously and without thought, to expand it into a coherent and 
amply informative dispatch. 

This expansion of the sensational core to produce what goes 
by the somewhat question-begging name of “perception” is 
obviously only trustworthy in so far as our habits of association 
run parallel to processes in the external world. Clouds looked 
down upon from a mountain may look so like the sea or a field of 
snow that only positive knowledge to the contrary prevents you 
from so interpreting your visual sensations. If you are not accus- 
tomed to the gramophone, you will confidently believe that the 
voice you hear on the other side of the door proceeds from a 
person in the room that you are about to enter. There is no 
obvious limit to the invention of ingenious apparatus capable of 
deceiving the unwary. We know that the people we see on the 
screen in the cinema are not really there, although they move and 
talk and behave in a manner having some resemblance to that of 
human beings; but if we did not know it, we might at first find 
it hard to believe. Thus what we seem to know through the senses 
may be deceptive whenever the environment is different from 
what our past experience has led us to expect. 

From the above considerations it follows that we cannot admit 
as data all that an uncritical acceptance of common sense would 
take as given in perception. Only sensations and memories are 
truly data for our knowledge of the external world. We must 
exclude from our list of data not only the things that we con- 
sciously infer, but all that is obtained by animal inference, such 
as the imagined hardness of an object seen but not touched. It is 
true that our “perceptions”, in all their fullness, are data for 
psychology: we do in fact have the experience of believing in 
such-and-such an object. It is only for knowledge of things outside 
our own minds that it is necessary to regard only sensations as 
data. This necessity is a consequence of what we know of physics 
and physiology. The same external stimulus, reaching the brains 
of two men with different experiences, will produce different 
results, and it is only what these different results have in common 
that can be used in inferring external causes. If it is objected that 
the truth of physics and physiology is doubtful, the situation is 
even worse; for if they are false, nothing whatever as to the outer 
world can be inferred from my experiences. I am, however, 

human knowledge: its scope and limits 

throughout this work, assuming that science is broadly speaking 

If we define “data” as “those matters of fact of which, inde- 
pendently of inference, we have a right to feel most nearly certain”, 
it follows from what has been said that all my data are events that 
happen to me, and are, in fact, what would commonly be called 
events in my mind. This is a view which has been characteristic 
of British empiricism, but has been rejected by most Continental 
philosophers, and is not now accepted by the followers of Dewey 
or by most of the logical positivists. As the issue is of considerable 
importance, I shall set forth the reasons which have convinced 
me, including a brief repetition of those that have already been 

There are, first, arguments on the common-sense level, derived 
from illusions, squinting, reflection, refraction, etc., but above all 
from dreams. I dreamed last night that I was in Germany, in a 
house which looked out on a ruined church; in my dream I 
supposed at first that the church had been bombed during the 
recent war, but was subsequently informed that its destruction 
dated from the wars of religion in the sixteenth century. All this, 
so long as I remained asleep, had all the convincingness of waking 
life. I did really have the dream, and did really have an experience 
intrinsically indistinguishable from that of seeing a ruined church 
when awake. It follows that the experience which I call “seeing a 
church” is not conclusive evidence that there is a church, since it 
may occur when there is no such external object as I suppose in 
my dream. It may be said that, though when dreaming I may 
think that I am awake, when I wake up I know that I am awake. 
But I do not see how we are to have any such certainty; I have 
frequently dreamt that I woke up; in fact once, after ether, I 
dreamt it about a hundred times in the course of one dream. We 
condemn dreams, in fact, because they do not fit into a proper 
context, but this argument can be made inconclusive, as in 
Calderon’s play, La Vida es Sueno. I do not believe that I am 
now dreaming, but I cannot prove that I am not. I am, however, 
quite certain that I am having certain experiences, whether they 
be those of a dream or those of waking life. 

We come now to another class of arguments, derived from 
physics and physiology. This class of arguments came into 
philosophy with Locke, who used it to show that secondary 

1 86 


qualities are subjective. This class of arguments is capable of 
being used to throw doubt on the truth of physics and physiology, 
but I will deal with them on the hypothesis that science, in the 
main, is true. 

We experience a visual sensation when light waves reach the 
eye, and an auditory sensation when sound waves reach the ear. 
There is no reason to suppose that light waves are at all like the 
experience which we call seeing something, or sound waves at all 
like the experience which we call hearing a sound. There is no 
reason whatever to suppose that the physical sources of light and 
sound waves have any more resemblance to our experiences than 
the waves have. If the waves are produced in unusual ways, our 
experience may lead us to infer subsequent experiences which it 
turns out that we do not have; this shows that even in normal 
perception interpretation plays a larger part than common sense 
supposes, and that interpretation sometimes leads us to entertain 
false expectations. 

Another difficulty is connected with time. We see and hear 
now, but what (according to common sense) we are seeing and 
hearing occurred some time ago. When we both see and hear an 
explosion, we see it first and hear it afterwards. Even if we 
could suppose that the furniture of our room is exactly what 
it seems, we cannot suppose this of a nebula millions of light- 
years away, which looks like a speck but is not much smaller 
than the milky way, and of which the light that reaches us 
now started before human beings began to exist. And the 
difference between the nebula and the furniture is only one of 

Then there are physiological arguments. People who have lost 
a leg may continue to feel pain in it. Dr. Johnson, disproving 
Berkeley, thought the pain in his toe when he kicked a stone was 
evidence for the existence of the stone, but it appears that it was 
not even evidence for the existence of his toe, since he might have 
felt it even if his toe had been amputated. Speaking generally, 
if a nerve is stimulated in a given manner, a certain sensation 
results, whatever may be the source of the stimulation. Given 
sufficient skill, it ought to be possible to make a man see the 
starry heavens by tickling his optic nerve, but the instrument used 
would bear little resemblance to the august bodies studied by 


human knowledge: its scope and limits 

The above arguments, as I remarked before, may be interpreted 
sceptically, as showing that there is no reason to believe that our 
sensations have external causes. As this interpretation concedes 
what I am at present engaged in maintaining, namely that sensa- 
tions are the sole data for physics, I shall not, for the moment, 
consider whether it can be refuted, but shall pass on to a closely 
similar line of argument which is related to the method of 
Cartesian doubt. This method consists in searching for data by 
provisionally rejecting everything that it is found possible to call 
in question. 

Descartes argues that the existence of sensible objects might be 
uncertain, because it would be possible for a deceitful demon to 
mislead us. We should substitute for a deceitful demon a cinema 
in technicolour. It is, of course, also possible that we may be 
dreaming. But he regards the existence of our thoughts as wholly 
unquestionable. When he says “I think, therefore I am”, the 
primitive certainties at which he may be supposed to have arrived 
are particular “ though ts”, in the large sense in which he uses the 
term. His own existence is an inference from his thoughts, an 
inference whose validity does not at the moment concern us. In 
the context, what appears certain to him is that there is doubting, 
but the experience of doubting has no special prerogative over 
other experiences. When I see a flash of lightning I may, it is 
maintained, be uncertain as to the physical character of lightning, 
and even as to whether anything external to myself has happened, 
but I cannot make myself doubt that there has been the occurrence 
which is called “seeing a flash of lightning”, though there may 
have been no flash outside my seeing. 

It is not suggested that I am certain about all my own experi- 
ences; this would certainly be false. Many memories are dubious, 
and so are many faint sensations. What I am saying — and in this 
I am expounding part of Descartes’ argument — is that there are 
some occurrences that I cannot make myself doubt, and that these 
are all of the kind that, if we admit a not-self, are part of the life 
of my self. Not all of them are sensations; some are abstract 
thoughts, some are memories, some are wishes, some are pleasures 
or pains. But all are what we should commonly describe as mental 
events in me. 

My own view is that this point of view is in the right in so far 
as it is concerned with data that are matters of fact. Matters of 

1 88 


tact that lie outside my experience can be made to seem doubtful, 
unless there is an argument showing that their existence follows 
from matters of fact within my experience together with laws of 
whose certainty I feel reasonably convinced. But this is a long 
question, concerning which, at the moment, I wish to say only a 
few preliminary words. 

Hume’s scepticism with regard to the world of science resulted 
from [a) the doctrine that all my data are private to me, together 
with ( b ) the discovery that matters of fact, however numerous and 
well-selected, never logically imply any other matter of fact. I 
do not see any way of escaping from either of these theses. The 
first I have been arguing; I may say that I attach especial weight 
in this respect to the argument from the physical causation of 
sensations. As to the second, it is obvious as a matter of syntax to 
any one who has grasped the nature of deductive arguments. A 
matter of fact which is not contained in the premisses must 
require for its assertion a proper name which does not occur in 
the premisses. But there is only one way in which a new proper 
name can occur in a deductive argument, and that is when we 
proceed from the general to the particular, as m “all men are 
mortal, therefore Socrates is mortar’. Now no collection of 
assertions of matters of fact is logically equivalent to a general 
assertion, so that, if our premisses concern only matters of fact, 
this way of introducing a new proper name is not open to us. 
Hence the thesis follows. 

If we are not to deduce Ilume’s scepticism from the above two 
premisses, there seems to be only one possible way of escape, 
and that is to maintain that, among the premisses of our knowledge, 
there are some general propositions, or there is at least one general 
proposition, which is not analytically necessary, i.e. the hypothesis 
of its falsehood is not self-contradictory. A principle justifying the 
scientific use of induction would have this character. What is 
needed is some way of giving probability (not certainty) to the 
inferences from known matters of fact to occurrences which have 
not yet been, and perhaps never will be, part of the experience 
of the person making the inference. If an individual is to know 
anything beyond his own experiences up to the present moment, 
his stock of uninferred knowledge must consist not .only of 
matters of fact, but also of general laws, or at least a law, allowing 
him to make inferences from matters of fact; and such law or 


human knowledge: its scope and limits 

laws must, unlike the principles of deductive logic, be synthetic, 
i.e. not proved true by their falsehood being self-contradictory. 
The only alternative to this hypothesis is complete scepticism as 
to all the inferences of science and common sense, including those 
which I have called animal inferences. 


Chapter II 

T he doctrine called “solipsism” is usually defined as the 
belief that I alone exist. It is not one doctrine unless it is 
true. If it is true, it is the assertion that 1, Bertrand Russell, 
alone exist. But if it is false, and I have readers, then for you who 
are reading this chapter it is the assertion that you alone exist. 
This is a view suggested by the conclusions reached in the 
preceding chapter, to the effect that all my data, in so far as they 
are matters of fact, are private to me, and that inferences from 
one or more matters of fact to other matters of fact are never 
logically demonstrative. These conclusions suggest that it would 
be rational to doubt everything outside my own experience, such 
as the thoughts of other people and the existence of material 
objects when I am not seeing them. It is this view that we are 
now to examine. 

We must begin by giving more precision to the doctrine, and 
by distinguishing various forms that it may take. We must not 
state it in the words “1 alone exist”, for these words have no clear 
meaning unless the doctrine is false. If the world is really the 
common-sense world of people and things, we can pick out one 
person and suppose him to think that he is the whole universe. 
This is analogous to the people before Columbus, who believed 
the Old World to be the total of land on this planet. But if other 
people and things do not exist, the word “myself” loses its mean- 
ing, for this is an exclusive and delimiting word. Instead of saying 
“myself is the whole universe”, we must say “data are the whole 
universe”. Here “data” may be defined by enumeration. We can 
then say: “this list is complete; there is nothing more”. Or we 
can say: “there is not known to be anything more”. In this form, 
the doctrine does not require a prior definition of the Self, and 
what it asserts is sufficiently definite to be discussed. 

(We may distinguish two kinds of solipsism, which I shall call 
“dogmatic” and “sceptical” respectively. The dogmatic kind, in 
the above statement, says “there is nothing beyond data”, while 
the sceptical kind says “there is not known to be anything beyond 
data”. No grounds exist in favour of the dogmatic form, since it 


HUMAN knowledge: its scope and limits 

is just as difficult to disprove existence as to prove it, when what 
is concerned is something which is not a datumi I shall therefore 
say no more about dogmatic solipsism, and shall concentrate on 
the sceptical form. 

The sceptical form of the doctrine is difficult to state precisely. 
It is not right to say, as we did just now, “nothing is known 
except data”, since some one else might know more; there is the 
same objection as there is to dogmatic solipsism. If we emend our 
statement by saying “nothing is known to me except the following 
(giving a list of data)”, we have again introduced the Self, which, 
as we saw, we must not do in defining our doctrine. It is not 
altogether easy to evade this objection. 

I think we can state the problem with which solipsism is 
concerned as follows: “The propositions p x , /> 2 , . . . p n are known 
otherwise than by inference. Can this list be made such that from 
it other propositions, asserting matters of fact, can be inferred?” 
In this form we do not have to state that our list is complete, or 
that it embraces all that some one person knows. 

It is obvious that if our list consists entirely of propositions 
asserting matters of fact, then the answer to our question is in 
the negative, and sceptical solipsism is true. But if our list contains 
anything in the nature of laws the answer may be different. These 
laws, however, will have to be synthetic. Any collection of matters 
of fact is logically capable of being the whole ; in pure logic, any 
two events are compossible, and no collection of events implies 
the existence of other events. 

But before pursuing this line of thought let us consider different 
forms of solipsism. 

Solipsism may be more drastic or less drastic; as it becomes 
more drastic it becomes more logical and at the same time more 
unplausible. In its least drastic form, it accepts all my mental 
states that are accepted by common sense or by orthodox 
psychology, i.e. not only those of which I am directly aware, but 
also those that are inferred on purely psychological grounds. It is 
generally held that at all times I have many faint sensations that 
I do not notice. If there is a ticking clock in the room, I may 
notice it and be annoyed by it, but as a rule I am quite unaware 
of it, even if it is easily audible whenever I choose to listen to it. 
In such a case one would naturally say that I am having auditory 
sensations of which I am not conscious. The same may be said, 



at most times, of objects in the periphery of my field of vision. 
If they are important objects, such as an enemy with a loaded 
revolver, I shall quickly become aware of them and bring them 
into the centre of my visual field : but if they are uninteresting and 
motionless I shall remain unaware of them. Nevertheless it seems 
natural to suppose that I am in some sense seeing them. 

The same sort of considerations apply to lapses of memory. 
If I look at an old diary, I find dinner engagements noted that I 
have completely forgotten, but I find it hard to doubt that I had 
the experience which common sense would describe as going to 
a dinner party. I believe that I was once an infant, although no 
trace of that period survives in my explicit memory. 

Such inferred mental states are allowed by the least drastic 
form of solipsism. It merely refuses to allow inferences to any- 
thing other than myself and my mental states. 

This, however, is illogical. The principles required to justify 
inferences from mental states of which I am aware to others of 
which I am not aware are exactly the same as those required for 
inferences to physical objects and to other minds. If, therefore, 
we are to secure the logical safety of which solipsism is in search, 
we must confine ourselves to mental states of which we are now 
aware. Buddha was admired because he could meditate while 
tigers roared around him ; but if he had been a consistent solipsist 
he would have held that the noise of roaring ceased as soon as he 
ceased to notice it. 

We thus arrive at a second form of solipsism, which says that 
the universe consists, or perhaps consists, of only the following 
items; and then we enumerate whatever, at the moment of 
speaking, we perceive or remember. And this will have to be 
confined to what I actually notice, for what I could notice is 
inferred. At the moment, I notice my dog asleep, and as a plain 
man I am convinced that I could have noticed him any time this 
last hour, since he has been consistently (so I believe) in my field 
of vision, but I have in fact been quite unaware of him. The 
thoroughgoing solipsist will have to say that when, during the 
last hour, my eye absent-mindedly rested on the dog, nothing 
whatever occurred in me in consequence ; for to argue that I had 
a sensation which I did not notice is to allow an inference of the 
forbidden kind. 

In regard to memory, the results of this theory are extremely 

193 N 

HUMAN knowledge: its scope and limits 

odd. The things that I am recollecting at one moment are quite 
different from those that I am recollecting at another, but the 
thoroughgoing solipsist should only admit what I am remember- 
ing now. Thus his world will be one of disjointed fragments 
which change completely from moment to moment — change, I 
mean, not as to what exists now, but as to what did exist in the 

But we have not done with the sacrifices which the solipsist 
must make to logic if he is to feel safe. It is quite clear that I can 
have a recollection without the thing remembered having hap- 
pened; as a matter of logical possibility, I might have begun to 
exist five minutes ago, complete with all the memories that I then 
had. We ought therefore to cut out events remembered, and 
confine the solipsist’s universe to present percepts, including 
percepts of present states of mind which purport to be recollec- 
tions. With regard to present percepts, this most rigorous type 
of solipsist (if he exists) accepts the premiss of Descartes’ cogito> 
with some interpretation. What he admits can only be correctly 
stated in the form: “A, B, C, . . . occur.” To call A, B, C, . . . 
“thoughts” adds nothing except for those who reject solipsism. 
What distinguishes the consistent solipsist is the fact that the 
proposition “A occurs”, if it comes in his list, is never inferred. 
He rejects as invalid all inferences from one or more propositions 
of the form “A occurs” to other propositions asserting the 
occurrence of something, whether named or described. The 
conclusions of such inferences, he maintains, may or may not 
happen to be true, but can never be known to be true. 

Having now stated the solipsist position, we must inquire what 
can be said for and against it. 

The argument for sceptical solipsism is as follows: From a 
group of propositions of the form “A occurs”, it is impossible to 
infer by deductive logic any other proposition asserting the 
occurrence of something. If any such inference is to be valid, it 
must depend upon some non-deductive principle such as causality 
or induction. No such principle can be shown to be even probable 
by means of deductive arguments from a group of propositions 
of the form “A occurs”. (I shall be concerned in a later chapter 
with the proof of this assertion.) For example, the validity of 
induction cannot be inferred from the course of events except by 
assuming induction or some equally questionable postulate. 



Therefore if, as empiricists maintain, all our knowledge is based 
on experience, it must be not only based on experience, but 
confined to experience ; for it is only by assuming some principle 
or principles which experience cannot render even probable that 
anything whatever can be proved by experience except the 
experience itself. 

I think this argument proves that we have to choose between 
two alternatives. Either we must accept sceptical solipsism in its 
most rigorous form, or we must admit that we know, indepen- 
dently of experience, some principle or principles by means of 
which it is possible to infer events from other events, at least with 
probability. If we adopt the first alternative, we must reject far 
more than solipsism is ordinarily thought to reject; we cannot 
know of the existence of our own past or future, or have any 
ground for expectations as to our own future, if it occurs. If we 
adopt the second alternative, we must partially reject empiricism; 
we must admit that we have knowledge as to certain general 
features of the course of nature, and that this knowledge, though 
it may be caused by experience, cannot be logically inferred from 
experience. We must admit also that, if we have such knowledge, 
it is not yet explicit; causality and induction, in their traditional 
forms, cannot be quite true, and it is by no means clear what 
should be substituted for them. It thus appears that there are 
great difficulties in the way of accepting either alternative. 

For my part, I reject the solipsist alternative and adopt the 
other. I admit, what is of the essence of the matter, that the 
solipsist alternative cannot be disproved by means of deductive 
arguments, provided we grant what I shall call “the empiricist 
hypothesis”, namely that what we know without inference consists 
solely of what we have experienced (or, more strictly, what we are 
experiencing) together with the principles of deductive logic. 
But we cannot know the empiricist hypothesis to be true, since 
that would be knowledge of a sort that the hypothesis itself 
condemns. This does not prove the hypothesis to be false, but it 
does prove that we have no right to assert it. Empiricism may be 
a true philosophy, but if it is it cannot be known to be true ; those 
who assert that they know it to be true contradict themselves. 
There is therefore no obstacle ab initio to our rejecting the 
empiricist hypothesis. 

As against solipsism it is to be said, in the first place, that it is 


human knowledge: its scope and limits 

psychologically impossible to believe, and is rejected in fact even 
by those who mean to accept it. I once received a letter from an 
eminent logician, Mrs. Christine Ladd Franklin, saying that she 
was a solipsist, and was surprised that there were no others. 
Coming from a logician, this surprise surprised me. The fact that 
I cannot believe something does not prove that it is false, but it 
does prove that I am insincere and frivolous if I pretend to believe 
it. Cartesian doubt has value as a means of articulating our 
knowledge and showing what depends on what, but if carried too 
far it becomes a mere technical game in which philosophy loses 
seriousness. Whatever anybody, even I myself, may argue to the 
contrary, I shall continue to believe that I am not the whole 
universe, and in this every one will in fact agree with me, if I am 
right in my conviction that other people exist. 

The most important part of the argument as to solipsism is the 
proof that it is only tenable in its most drastic form. There are 
various half-way positions which are not altogether unplausible, 
and have in fact been accepted by many philosophers. Of these 
the least drastic is the view that there can never be good grounds 
for asserting the existence of something which no one experiences ; 
from this we may, with Berkeley, infer the unreality of matter 
while retaining the reality of mind. But this view, since it admits 
the experiences of others than myself, and since these experiences 
are only known to me by inference, considers that it is possible 
to argue validly from the existence of certain occurrences to the 
existence of others; and if this is admitted, it will be found that 
there is no reason why the inferred events should be experienced. 
Exactly similar considerations apply to the form of solipsism 
which believes that oneself has a past and a probable future ; this 
belief can only be justified by admitting principles of inference 
which lead to the rejection of every form of solipsism. 

We are thus reduced to the two extreme hypotheses as alone 
logically defensible. Either, on the one hand, we know principles 
of non-deductive inference which justify our belief, not only in 
other people, but in the whole physical world, including the parts 
which are never perceived but only inferred from their effects; 
or, on the other hand, we are confined to what may be called 
“solipsism of the moment”, in which the whole of my knowledge 
is limited to what I am now noticing, to the exclusion of my past 
and probable future, and also of all those sensations to which, at 



this instant, I am not paying attention. When this alternative is 
clearly realized, I do not think that anybody would honestly and 
sincerely choose the second hypothesis. 

If solipsism of the moment is rejected, we must seek to dis- 
cover what are the synthetic principles of inference by the know- 
ledge of which our scientific and common-sense beliefs are to be 
justified in their broad outlines. To this task we shall address 
ourselves in Part VI. But it will be well first to make a survey, on 
the one hand of data, and on the other hand of scientific beliefs 
interpreted in their least questionable form. By analysing the 
results of this survey we may hope to discover the premisses 
which, consciously or unconsciously, are assumed in the reasonings 
of science. 


Chapter III 


A “probable” inference (to repeat what has already been 
said) is one in which, when the premisses are true and the 
reasoning correct, the conclusion is nevertheless not 
certain but only probable in a greater or less degree. In the 
practice of science there are two kinds of inferences: those that 
are purely mathematical, and those that may be called “sub- 
stantial”. The inference from Kepler’s laws to the law of gravita- 
tion as applied to the planets is mathematical, but the inference 
from the recorded apparent motions of the planets to Kepler’s 
laws is substantial, for Kepler’s laws are not the only hypotheses 
logically compatible with observed facts. Mathematical inference 
has been sufficiently investigated during the last half-century. 
What I wish to discuss is non-mathematical inference, which is 
always only probable. 

I shall, broadly speaking, accept as valid any inference which 
is part of the accepted body of scientific theory, unless it contains 
some error of a specific kind. I shall not consider the arguments 
for scepticism concerning science, but shall analyse scientific 
inference on the hypothesis that it is in general valid. 

My concern in this chapter will be mainly with pre-scientific 
knowledge as embodied in common sense. 

We must bear in mind the distinction between inference as 
understood in logic and what may be called “animal inference”. 
By “animal inference” I mean what happens when an occurrence 
A causes a belief B without any conscious intermediary. When a 
dog smells a fox he becomes excited, but we do not think that he 
says to himself: “This smell has in the past been frequently 
associated with the neighbourhood of a fox; therefore there is 
probably a fox in the neighbourhood now.” He acts, it is true, 
as he would if he went through this reasoning, but the reasoning 
is performed by the body, through habit, or the “conditioned 
reflex” as it is called. Whenever A has, in the animal’s past 
experience, been frequently associated with B, where B is some- 
thing of emotional interest, the occurrence of A tends to cause 



behaviour appropriate to B. There is here no conscious connection 
of A and B ; there is, we may say, A-perception and B-behaviour. 
In old-fashioned language, it would be said that the “impression” 
of A causes the “idea” of B. But the newer phraseology, in terms 
of bodily behaviour and observable habit, is more precise and 
covers a wider field. 

Most substantial inferences in science, as opposed to merely 
mathematical inferences, arise, in the first place, from analysis 
of animal inferences. But before developing this aspect of our 
subject, let us consider the scope of animal inference in human 

The practical (as opposed to the theoretical) understanding of 
language comes under the head of animal inference. Understand- 
ing a word consists practically of (a) the effects of hearing it, and 
( b ) the causes of uttering it. You understand the word “fox” if, 
when you hear it, you have an impulse to act in a manner appro- 
priate to the presence of a fox, and when you see a fox, you have 
an impulse to say “fox”. But you do not need to be aware of this 
connection between foxes and the word “fox”; the inference from 
the word to the fox or from the fox to the word is an animal 
inference. It is otherwise with erudite words, such as “dodeca- 
hedron”. We learn the meaning of such words through a verbal 
definition, and in such cases the connection of word and meaning 
begins by being a conscious inference before it becomes a habit. 

Words are a particular case of signs. We may say that, to a 
given organism O, a member of a class of stimuli A is a sign 1 of 
some member of a class of objects B if the occurrence to O of a 
stimulus of class A produces a reaction appropriate to an object 
of class B. But this is not yet quite precise. Before seeking further 
precision, let us consider a concrete example, say “no smoke 
without fire”. 

There are various stages to be gone through before this proverb 
can be enunciated. First, there must be repeated experience of 
both smoke and fire, either simultaneously or in close temporal 
succession. Originally, each produced its own reaction, (say) 
smoke that of sniffing, and fire that of running away. But in time 
a habit is formed, and smoke produces the reaction of running 
away. (I am assuming an environment where forest fires are 
frequent.) Some ages after the first formation of this habit, two 
* Or, more correctly, a “subjective sign”. 


human knowledge: its scope and limits 

new habits are formed: smoke leads to the word “smoke”, and 
fire leads to the word “fire”. Where these three habits exist— 
smoke causing a reaction appropriate to fire, smoke causing the 
word “smoke” and fire causing the word “fire” — the materials 
exist for the formation of a fourth habit, that of the word “smoke” 
causing the word “fire”. When this habit exists in a reflective 
philosopher, it may cause the sentence “no smoke without fire”. 
Such, at least, is a bare outline of a very complex process. 

In the above example, when all these habits exist, smoke is a 
sign of fire, the word “smoke” is a sign of smoke, and the word 
“fire” is a sign of fire. Perhaps it may be assumed that the sign- 
relation is often transitive, i.e. that, if A is a sign of B and B is 
a sign of C, then A is a sign of C. This will not be invariably the 
case, but it will tend to happen if the sign-relations of A and B, 
B and C are very firmly established in the animal’s organism. In 
that case, when the word “smoke” is a sign of smoke, and smoke 
is a sign of fire, the word “smoke” will be, derivatively, a sign of 
fire. If fire causes the word “fire”, the word “smoke” will thus 
have become, derivatively, a cause of the word “fire”. 

Let us set up a definition : An organism O has an “idea” of a 
kind of object B when its action is appropriate to B although no 
object of the kind B is sensibly present. This, however, requires 
some limitation. An “idea” need not produce all the reactions 
that would be produced by the object; this is what we mean by 
saying that an idea may be faint, or not vividly imagined. There 
may be nothing but the word “B”. Thus we shall say that the 
idea of B is present to O whenever O shows some reaction appro- 
priate to B and to nothing else. 

We can now say that A is a sign of B when A causes the “idea” 
of B. 

We have used the word “appropriate”, and this word needs 
further definition. It must not be defined teleologically, as “useful 
to the organism” or what not. The reaction “appropriate” to B 
is primarily the reaction caused by the sensible presence of B, 
independently of acquired habits. A cry of pain on contact with 
something very hot is an appropriate reaction in this sense. But 
we cannot altogether exclude acquired habits from our definition 
of appropriate reactions. To say “fox” when you see a fox is 
appropriate. We may make a distinction: there is no situation to 
which, apart from acquired habits, we react by saying “fox”. 



We may therefore decide to include among “appropriate** re- 
actions those which, as a result of habit, occur in the presence of 
the object B, but do not occur spontaneously as reactions to 
anything except B, and do not occur as habitual reactions to any- 
thing other than B except as a result of a combination of habits. 

The above discussion gives the definition of what may be 
called a “subjective** sign, when A causes the idea of B. We may 
say that A is an “objective** sign of B when A is in fact followed 
or accompanied by B, and not only by the idea of B. We may say 
roughly that there is error on the part of an organism whenever 
a subjective sign is not also an objective sign ; but such a statement 
is not correct without qualification. 

Qualification is required because we must distinguish an idea 
accompanied by belief from one merely entertained. If you had 
two friends called Box and Cox, it is probable that the sight of 
Box would cause the idea of Cox, but not the belief in the 
presence of Cox. I think that entertaining an idea without belief 
is a more complex occurrence than entertaining it with belief. 
An idea is or involves (I will not argue which) an impulse to a 
certain kind of action. When the impulse is uninhibited, the idea 
is “believed”; when inhibited, the idea is merely “entertained**. 
In the former case we may call the idea “active**, in the latter 
“suspended**. Error is only connected with active ideas. Thus 
there is error when a subjective sign produces an active idea, 
although there is no such sequence between the sign and the 
object of the idea. 

Error, according to this view, is pre-intellectual; it requires 
only bodily habits. There is error when a bird flies against a pane 
of glass which it does not see. We all, like the bird, entertain rash 
beliefs which may, if erroneous, lead to painful shocks. Scientific 
method, I suggest, consists mainly in eliminating those beliefs 
which there is positive reason to think a source of shocks, while 
retaining those against which no definite argument can be brought. 

In what I have been saying I have been assuming causal laws of 
the form “A causes B”, where A and B are classes of occurrences. 
Such laws are perhaps never wholly true. True laws can only be 
expressed in differential equations. But it is not necessary that 
they should be exactly true. What we need is only: “In a good 
deal more than half the cases in which A occurs, B occurs simul- 
taneously or soon afterwards.** This makes B probable whenever 


HUMAN knowledge: its scope and limits 

A has occurred, and that is as much as we should demand. I have 
assumed that if, in the history of a given organism, A has often 
been followed by B, A will be accompanied or quickly followed by 
the “idea” of B, i.e. by an impulse to the actions which would be 
stimulated by B. This law is inevitably vague. If A and B are 
emotionally interesting to the organism, one case of their con- 
junction may suffice to set up a habit; if not, many may be needed. 
The conjunction of 54 and 6 times 9 has, for most children, little 
emotional interest; hence the difficulty of learning the multi- 
plication table. On the other hand, “once bit twice shy” shows 
how easily a habit is formed when the emotional interest is strong. 

As appears from what we have been saying, science starts, and 
must start, from rough and ready generalizations which are only 
approximately true, many of which exist as animal inferences 
before they are put into words. The process is as follows: A is 
followed by B a certain number of times ; then A is accompanied 
by the expectation of B; then (probably much later) comes the 
explicit judgment “A is a sign of B”; and only then, when 
multitudes of such judgments already exist, can science begin. 
Then comes Hume, with his query as to whether we ever have 
reason to regard A as an objective sign of B, or even to suppose 
that we shall continue to think it a sign of B. This is a sketch of 
the psychology of the subject; it has no direct bearing on its logic. 

The distinction between animal inference and scientific infer- 
ence, I repeat, is this : In animal inference, the percept A causes 
the idea of B, but there is no awareness of the connection; in 
scientific inference (whether valid or invalid) there is a belief 
involving both A and B, which I have expressed by “A is a sign 
of B”. It is the occurrence of a single belief expressing a con- 
nection of A and B that distinguishes what is commonly called 
inference from what I call animal inference. But it is important 
to notice that the belief expressing the connection is, in all the 
most elementary cases, preceded by the habit of animal inference. 

Take, as an example, the belief in more or less permanent 
objects. A dog, seeing his master on different occasions, reacts in 
a way which has some constant features; this is the observable 
fact which we express by saying that the dog “recognizes” his 
master. When the dog looks for his absent master, something 
more than recognition is involved. It is difficult not to use unduly 
intellectualist language to describe what occurs in such a case. 



One might be tempted to say that there is a desire to replace an 
idea of an object by an impression of it, but this is the sort of 
phrase that seems to say much and really says little. The simplest 
observable fact about desire in animals is restless behaviour until 
a certain situation arises, and then relative quiescence. There are 
also physiological facts about the secretions of the glands, such as 
Pavlov employed. I am not denying that dogs have experiences 
more or less similar to those which we have when we feel desire, 
but this is an inference from their behaviour, not a datum. What 
we observe can be summed up by saying that a certain part of the 
dog’s behaviour is unified by reference to his master, as a planet’s 
behaviour is unified by reference to the sun. In the case of the 
planet, we do not infer that it “thinks” about the sun; in the case 
of the dog, most of us do make the corresponding inference. But 
that is a difference with which we need not be concerned as yet. 

When we come to language, it is natural to have a single word 
for those features of the environment which are connected 
together in the kind of way in which appearances of the dog’s 
master are connected together for the dog. Language has proper 
names for the objects with which we are most intimately associated, 
and general names for other objects. Proper names embody a 
common-sense metaphysic, which, as animal inference, antedates 
language. Consider such children’s questions as “Where’s 
mother?” “Where’s my ball?” These imply that mothers and 
balls, when not sensibly present, nevertheless exist somewhere, 
and can probably be made sensible by suitable action. This belief 
in permanent or quasi-permanent objects is based upon recog- 
nition, and thus implies memory in some sense. However that 
may be, it is clear that, by the time a child begins to speak, he 
has a habit of similar reactions to a certain group of stimuli, 
which, when reflected upon, becomes a belief in persistent 
common-sense objects. Much the same must have been true of 
mankind when developing language. The metaphysic of more or 
less permanent objects underlies the vocabulary and syntax of 
every language, and is the basis of the concept of substance. The 
only point about it that I am concerned to make at the moment is 
that it results from intellectualizing the animal inference involved 
in recognition. 

I come now to memory. What I wish to say about memory is, 
that its general though not invariable trustworthiness is a premiss 



of scientific knowledge, which is necessary if science is to be 
accepted as mainly true, but is not capable of being made even 
probable by arguments which do not assume memory. More 
precisely : When I remember something, it is probable that what 
I remember occurred, and I can form some estimate of the degree 
of probability by the vividness of my recollection. 

Let us first make clear what is meant logically by saying that 
memory is a premiss of knowledge. It would be a mistake to set 
up a general statement of the form: “What is remembered 
probably occurred.” It is rather each instance of memory that is 
a premiss. That is to say, we have beliefs about past occurrences 
which are not inferred from other beliefs, but which, nevertheless, 
we should not abandon except on very cogent grounds. (By 
“we”, here, I mean people versed in scientific method and careful 
as to what they believe.) The cogent grounds must necessarily 
involve one or more scientific laws, and also matters of fact, 
which may be either perceived or remembered. When Macbeth’s 
witches vanish, he doubts whether he ever saw them, because he 
believes in the persistence of material objects. But although any 
memory may come to be thought mistaken, it always has a certain 
weight, which makes us accept it in the absence of contrary 

A few words must be said at this point about scientific laws as 
opposed to particular facts. It is only by assuming laws that one 
fact can make another probable or improbable. If I remember 
that at noon yesterday I was in America, but five minutes earlier 
I was in Kamchatka, I shall think that one of my memories must 
be mistaken, because I am firmly persuaded that the journey 
cannot be performed in five minutes. But why do I think this? 
As an empiricist, I hold that laws of nature should be inferred 
inductively from particular facts. But how am I to establish 
particular facts about how long a journey has taken ? It is clear 
that I must rely partly on memory, since otherwise I shall not 
know that I have taken a journey. The ultimate evidence for any 
scientific law consists of particular facts, together with those 
principles of scientific inference which it is my purpose to in- 
vestigate. When I say that memory is a premiss, I mean that, 
among the facts upon which scientific laws are based, some are 
admitted solely because they are remembered. They are admitted, 
however, only as probable, and any one of them may be rejected 



later, after scientific laws have been discovered which make the 
particular memory improbable. But this improbability is only 
arrived at by assuming that most memories are veridical. 

The necessity of memory as a premiss may be made evident by 
asking the question : what reason have we for rejecting the hypo- 
thesis that the world came into existence five minutes ago ? If it 
had begun then just as, in fact, it then was, containing people 
with the habits and supposed memories that in fact people then 
had, there would be no possible way of finding out that they had 
only just begun to exist. Yet there is nothing logically impossible 
in the hypothesis. Nothing that is happening now logically implies 
anything that happened at another time. And the laws of nature 
by which we infer the past are themselves, as we have seen, 
dependent upon memories for the evidence in their favour. 
Consequently remembered facts must be included with perceived 
facts as part of our data, though we may as a rule assign a lower 
degree of credence to them than we do to facts of present per- 

There is a distinction to be made here, which is not without 
importance. A recollection is a present fact: I remember now what 
I did yesterday. When I say that memory is a premiss, I do not 
mean that from my present recollection I can infer the past event 
recollected. This may be in some sense true, but is not the 
important fact in this connection. The important fact is that the 
past occurrence is itself a premiss for my knowledge. It cannot 
be inferred from the present fact of my recollecting it except by 
assuming the general trustworthiness of memory, i.e. that an 
event remembered probably did take place. It is this that is the 
memory-premiss of knowledge. 

It must be understood that, when I say that this or that is a 
premiss, I do not mean that it is certainly true ; I mean only that 
it is something to be taken account of in arriving at the truth, 
but not itself inferred from something believed to be true. The 
situation is the same as that in a criminal trial in which the 
witnesses contradict each other. Each witness has a certain prima 
fade weight, and we have to seek a consistent system embracing 
as many of their statements as possible. 

I come now to another source of knowledge, namely testimony. 
I do not think that the general truthfulness of testimony needs to 
be a premiss in the finished structure of scientific knowledge, but 


HUMAN knowledge: its scope and limits 

it is a premiss in the early stages, and animal inference makes us 
prone to believe it. Moreover I think we shall find that, in the 
finished structure of science, there is a general premiss which is 
needed to secure the probable trustworthiness of testimony as 
well as certain other things. 

Let us first consider common-sense arguments, such as would 
have weight in a law court. If twelve people, each of whom lies as 
often as he speaks the truth, independently testify to a certain 
occurrence, the odds are 4095 to 1 that they are testifying truly. 
This may be taken as practical certainty, unless the twelve people 
all have a special motive for lying. This may happen. If two ships 
have a collision at sea, all the sailors on one ship swear one thing, 
and all the sailors on the other swear the opposite. If one of the 
ships has been sunk with all hands, there will be unanimous 
testimony, about which, nevertheless, lawyers experienced in 
such cases will feel sceptical. But we need not pursue such argu- 
ments, which are a matter for lawyers rather than philosophers. 

The common-sense practice is to accept testimony unless there 
is a positive reason against doing so in the particular case con- 
cerned. The cause, though not the justification, of this practice 
is the animal inference from a word or sentence to what it signifies. 
If you are engaged in a tiger hunt and somebody exclaims 
“tiger ” your body will, unless you inhibit your impulses, get 
into a state very similar to that in which it would be if you saw 
a tiger. Such a state is the belief that a tiger is in the neighbour- 
hood; thus you will be believing the testimony of the man who 
said “tiger”. The creation of such habits is half of learning the 
English language; the other half is the creation of the habit of 
saying “tiger ” when you see one. (I am omitting niceties of 
grammar and syntax.) You can, of course, learn to inhibit the 
impulse to belief ; you may come to know that your companion is 
a practical joker. But an inhibited impulse still exists, and if it 
ceased to exist you would cease to understand the word “tiger”. 
This applies even to such dry statements as “tigers are found in 
India and Eastern Asia”. You may think you hear this statement 
without any of the emotions appropriate to tigers, and yet it may 
cause during the following night a nightmare from which you 
wake in a cold sweat, showing that the impulses appropriate to 
the word “tiger” survived subconsciously. 

It is this primitive credulity about testimony which causes the 



success of advertising. Unless you are an unusually sophisticated 
person, you will, if you are told often and emphatically that 
so-and-so’s soaps or politics are the best, in the end come to 
believe it, with the result that so-and-so becomes a millionaire 
or a dictator as the case may be. However, I do not want to stray 
into politics, so I will say no more about this aspect of the belief 
in testimony. 

Testimony must be distinguished from information as to the 
meaning of a word, though the distinction is not always easy. 
You learn the correct use of the word “cat” because your parents 
say “cat” when you are noticing a cat. If they were not sufficiently 
truthful for this — if, when you are noticing a cat, they said some- 
times “dog”, sometimes “cow”, sometimes “crocodile” — you 
could never learn to speak correctly. The fact that we do learn to 
speak correctly is a testimonial to the habitual veracity of parents. 
But while, from the parent’s point of view, his utterance of the 
word “cat” is a statement, from the child’s point of view it is 
merely a step in the acquisition of language-habits. It is only after 
the child knows the meaning of the word “cat” that your utterance 
of the word is a statement for him as well as for you. 

Testimony is very important in one respect, namely, that it 
helps to build up the distinction between the comparatively public 
world of sense and the private world of thought, which is already 
well established when scientific thinking begins. I was once 
giving a lecture to a large audience when a cat stalked in and lay 
down at my feet. The behaviour of the audience persuaded me 
that I was not suffering from a hallucination. Some of our exper- 
iences, but not all, appear from the behaviour of others (including 
testimony) to be common to all who are in a certain neighbourhood 
and are making use of normal senses. Dreams have not this public 
character; no more do most “thoughts”. It must be noticed that 
the public character of (say) a clap of thunder is an inference, 
originally an animal inference. I hear thunder, and a person 
standing beside me says “thunder”. I infer that he heard thunder, 
and until I become a philosopher I make this inference with my 
body, i.e. my mind emerges believing that he heard the thunder 
without having gone through a “mental” process. When I become 
a philosopher I have to examine the body’s inferential propensities, 
including the belief in a public world which it has inferred from 
observing behaviour (especially speech- behaviour) similar to its own. 


HUMAN knowledge: its scope and limits 

From the point of view of the philosopher, the interesting 
question is not only whether the testimony you hear is intended 
to be truthful, but whether it has any intention of conveying 
information. There are here various stages towards meaningless- 
ness. When you hear an actor on the stage say “I have supped full 
with horrors”, you do not think he is complaining about rationing, 
and you know that his statements are not intended to be believed. 
When you hear a soprano voice on a phonograph lamenting her 
lover’s faithlessness in anguished tones, you know that there is 
no lady in the box, and that the lady who made the record was 
not expressing her own emotions, but only intending to give you 
pleasure in the contemplation of an imaginary sorrow. Then there 
was the Scotch ghost in the eighteenth century, which kept on 
repeating: ‘‘Once I was hap-hap-happy but noo I am mceserable”, 
which turned out to be a rusty spit. Lastly, there are the people 
in dreams, who say all sorts of things that, when we wake up, we 
are convinced nobody did say. 

For all these reasons, we cannot accept testimony at its face 
value. The question arises : why should we accept it at all ? 

We depend here, as we do when we believe in sound waves and 
light waves, upon an inference going beyond our experience. Why 
should not everything that seems to us to be testimony be like 
either the creakings of the rusty spit or the conversation of people 
in dreams? We cannot refute this hypothesis by reliance upon 
experience, for our experience may be exactly the same whether 
the hypothesis is true or false. And in any inference beyond future 
as well as past experience we cannot rely upon induction. In- 
duction argues that, if A has been frequently found to be followed 
by B, it will probably be found to be followed by B next time. 
This is a principle which remains entirely within experience, 
actual or possible. 

In the case of testimony, we depend upon analogy . The 
behaviour of other people’s bodies — and especially their speech 
behaviour — is noticeably similar to our own, and our own is 
noticeably associated with “mental” phenomena. (For the 
moment it does not matter what we mean by “mental”.) We 
therefore argue that other people’s behaviour is also associated 
with “mental” phenomena. Or rather, we accept this at first as 
an animal inference, and invent the analogy argument afterwards 
to rationalize the already existing belief. 



Analogy differs from induction — at least as I am using the 
words — by the fact that an analogical inference, when it passes 
outside experience, cannot be verified. We cannot enter into the 
minds of others to observe the thoughts and emotions which we 
infer from their behaviour. We must therefore accept analogy — 
in the sense in which it goes beyond experience — as an inde- 
pendent premiss of scientific knowledge, or else we must find 
some other equally effective principle. 

The principle of analogical inference will have to be more or 
less as follows : Given a class of cases in which A is accompanied 
or succeeded by B, and another class of cases in which it cannot 
be ascertained whether B is present or not, there is a probability 
(varying according to circumstances) that in these cases also B is 

This is not an accurate statement of the principle, which will 
need various limitations. But the necessary further refinements 
would not make much difference in relation to the problem with 
which we are concerned. 

A further step away from experience is involved in the inference 
to such things as sound waves and light waves. Let us concentrate 
on the former. Suppose at a point O, from which many roads 
radiate, you place a charge of gunpowder, and at a certain moment 
you cause it to explode. Every hundred yards along these roads 
you station an observer with a flag. A person in a stationary 
balloon observes all the observers, who have orders to wave their 
flags when they hear the noise of the gunpowder exploding. It is 
found that those who are equidistant from O all wave their flags 
at the same moment, while those who are further from O wave 
their flags later than those who are nearer; moreover, the time 
that elapses between the seen explosion and the waving of a given 
observer's flag is proportional to his distance from O. It is 
inferred by science (common sense concurring) that some process 
travels outward from O, and that, therefore, something connected 
with the sound is happening, not only where there are observers, 
but also where there are none. In this inference we pass outside 
all experience, not only outside our personal experience as in the 
case of testimony. We cannot therefore interpret science wholly 
in terms of experience, even when we include all experience. 

The principle used in the above inference may be called, 
provisionally, the principle of spatio-temporal continuity in 

209 o 

human knowledge: its scope and limits 

causal laws. This is the same thing as denial of action at a distance. 
We cannot believe that sounds arrive successively at successive 
observers unless something has travelled over the intervening 
space. If we deny this, our world becomes altogether too staccato 
to be credible. The basis of our belief, presumably, is the con- 
tinuity of all observed motions; thus perhaps analogy can be 
stretched to cover this inference. However, there is much to be 
said before we can be clear as to the principle governing such 
inferences. I therefore leave the further consideration of this 
subject for a later chapter. 

So far, I have been concerned in collecting rough and ready 
examples of elementary scientific inference. It remains to give 
precision to the results of our preliminary survey. 

I will end with a summary of the results of our present dis- 

When we begin to reflect, we find ourselves possessed of a 
number of habits which may be called “animal inferences’*. 
These habits consist of acting in the presence of A more or less 
as we should in the presence of B, and they result from the past 
conjunction of A and B in our experience. These habits, when we 
become conscious of them, cause such beliefs as “A is always (or 
usually) followed by B”. This is one of the main sources of the 
stock of beliefs with which we start when we begin to be scientific ; 
in particular, it includes the understanding of language. 

Another pre-scientific belief which survives in science is the 
belief in more or less permanent objects, such as people and things. 
The progress of science refines this belief, and in modern quantum 
theory not very much remains of it, but science could hardly have 
been created without it. 

The general, though not universal, trustworthiness of memory 
is an independent postulate. It is necessary to much of our know- 
ledge, and cannot be established by inference from anything that 
does not assume it. 

Testimony is, like memory, part of the sources of our primitive 
beliefs. But it need not itself be made into a premiss, since it can 
be merged in the wider premiss of analogy . 

Finally, to infer such things as sound waves and light waves 
we need a principle which may be called spatio-temporal causal 
continuity, or denial of action at a distance. But this last principle 
is complicated, and demands further discussion. 


Chapter IV 


T he question to be discussed in this Chapter is one which, 
in my opinion, has been far too little considered. It is this: 
Assuming physics to be broadly speaking true, can we 
know it to be true, and, if the answer is to be in the affirmative, 
does this involve knowledge of other truths besides those of 
physics? We might find that, if the world is such as physics says 
it is, no organism could know it to be such ; or that, if an organism 
can know it to be such, it must know some things other than 
physics, more particularly certain principles of probable inference. 

This question becomes acute through the problem of percep- 
tion. There have, from the earliest times, been two types of theory 
as to perception, one empirical, the other idealist. According to the 
empirical theory, some continuous chain of causation leads from 
the object to the percipient, and what is called “perceiving” the 
object is the last link in this chain, or rather the last before the 
chain begins to lead out of the percipient’s body instead of into 
it. According to the idealist theory, when a percipient happens 
to be in the neighbourhood of an object a divine illumination 
causes the percipient’s soul to have an experience which is like 
the object. 

Each of these theories has its difficulties. 

The idealist theory has its origin in Plato, but reaches its logical 
culmination in Leibniz, who held that the world consists of 
monads which never interact, but which all go through parallel 
developments, so that what happens to me at any instant has a 
similarity to what is happening to you at the same instant. When 
you think you move your arm, I think I see you moving it ; thus 
we are both deceived, and no one before Leibniz was sufficiently 
acute to unmask the deception, which he regards as the best proof 
of God’s goodness. This theory is fantastic, and has had few 
adherents ; but in less logical forms portions of the idealistic theory 
of perception are to be found even among those who think them- 
selves most remote from it. 

Philosophy is an offshoot of theology, and most philosophers, 
like Malvolio, “think nobly of the soul”. They are therefore pre- 

21 1 

HUMAN knowledge: its scope and limits 

disposed to endow it with magical powers, and to suppose that 
the relation between perceiving and what is perceived must be 
something utterly different from physical causation. This view 
is reinforced by the belief that mind and matter are completely 
disparate, and that perceiving, which is a mental phenomenon, 
must be totally unlike an occurrence in the brain, which is all 
that can be attributed to physical causation. 

The theory that perceiving depends upon a chain of physical 
causation is apt to be supplemented by a belief that to every state 
of the brain a certain state of the mind “corresponds”, and vice 
versa, so that, given either the state of the brain or the state of the 
mind, the other could be inferred by a person who sufficiently 
understood the correspondence. If it is held that there is no causal 
interaction between mind and brain, this is merely a new form of 
the pre-established harmony. But if causation is regarded — as it 
usually is by empiricists — as nothing but invariable sequence or 
concomitance, then the supposed correspondence of brain and 
mind tautologically involves causal interaction. The whole 
question of the dependence of mind on body or body on mind 
has been involved in quite needless obscurity owing to the emo- 
tions involved. The facts are quite plain. Certain observable 
occurrences are commonly called “physical”, certain others 
“mental”; sometimes “physical” occurrences appear as causes of 
“mental” ones, sometimes vice versa. A blow causes me to feel 
pain, a volition causes me to move my arm. There is no reason to 
question either of these causal connections, or at any rate no 
reason which does not apply to all causal connections equally. 

These considerations remove one set of difficulties that stand 
in the way of acceptance of the physical theory of perception. 

The common-sense arguments in favour of the physical causa- 
tion of perceptions are so strong that only powerful prejudices 
could have caused them to be questioned. When we shut our 
eyes we do not see, when we stop our ears we do not hear, when 
we are under an anaesthetic we perceive nothing. The appearance 
that a thing presents can be altered by jaundice, short sight, 
microscopes, mists, etc. The time at which we hear a sound 
depends upon our distance from its physical point of origin. The 
same is true of what we see, though the velocity of light is so 
great that, where terrestrial objects are concerned, the time 
between an occurrence and our seeing of it is inappreciable. If it 



is by a divine illumination that we perceive objects, it must be 
admitted that the illumination adapts itself to physical conditions. 

There are, however, two objections to the physical causation 
of perceptions. One is that it makes it impossible, or at least very 
difficult, to suppose that external objects are what they seemto 
be; the other is that it seems to make it doubtful whether the 
occurrences that we call “perceptions” can really be a source of 
knowledge as to the physical world. The first of these may be 
ignored as having only to do with prejudices, but the second is 
more important. 

The problem is this : Every empiricist holds that our knowledge 
as to matters of fact is derived from perception, but if physics is 
true there must be so little resemblance between our percepts 
and their external causes that it is difficult to see how, from per- 
cepts, we can acquire a knowledge of external objects. The 
problem is further complicated by the fact that physics has been 
inferred from perception. Historically, physicists started from 
naive realism, that is to say, from the belief that external objects 
are exactly as they seem; on the basis of this assumption, they 
developed a theory which made matter something quite unlike 
what we perceive. Thus their conclusion contradicted their 
premiss, though no one except a few philosophers noticed this. 
We therefore have to decide whether, if physics is true, the hypo- 
thesis of naive realism can be so modified that there shall be a 
valid inference from percepts to physics. In a word : If physics is 
true, is it possible that it should be known ? 

Let us first try to define what we are to mean by the hypothesis 
that physics is true. I want to adopt this hypothesis only to the 
extent to which it appeals to educated common sense. We find 
that the theories of physicists constantly undergo modification, 
so that no prudent man of science would expect any physical 
theory to be quite unchanged a hundred years hence. But when 
theories change, the alteration usually has only a small effect so 
far as observable phenomena are concerned. The practical differ- 
ence between Einstein's theory of gravitation and Newton’s is 
very minute, though the theoretical difference is very great. 
Moreover, in every new theory there are some parts that seem 
pretty certain, while others remain very speculative. Einstein’s 
substitution of space-time for space and time represents a change 
of language for which there are the same sort of grounds of 




simplicity as there were for the Copemican change of language. 
This part of Einstein’s theory may be accepted with considerable 
confidence. But the view that the universe is a three-dimensional 
sphere of finite diameter remains speculative; no one would be 
surprised if evidence were found which would lead astronomers 
to give up this way of speaking. 

Or, again, take the physical theory of light. No one doubts that 
light travels at the rate of roughly 300,000 kilometers per second, 
but whether it consists of waves, or of particles called photons, is 
a matter as to which dispute has been possible. In the case of 
sound, on the other hand, the wave theory may be accepted as 
firmly established. 

Every physical theory which survives goes through three stages. 
In the first stage, it is a matter of controversy among specialists; 
in the second stage, the specialists are agreed that it is the theory 
which best fits the available evidence, though it may well here- 
after be found incompatible with new evidence; in the third 
stage, it is thought very unlikely that any new evidence will do 
more than somewhat modify it. 

When I say that I shall assume physics to be true, I mean that 
I shall accept those parts of physics which have reached the third 
stage, not as certain, but as more probable than any philosophical 
speculation, and therefore proper to be accepted by philosophers 
as a premiss in their arguments. 

Let us now see what the most certain parts of physics have to 
say that is relevant to our present problem. 

The great physical discoveries of the seventeenth century were 
made by means of two working hypotheses. One of these was that 
causal laws in the physical world need only take account of matter 
and motion, matter being composed of particles persisting through 
time but continuously changing their positions in space. It was 
assumed that, so far as physics is concerned, there is no need to 
take account of anything about a particle except its position in 
space at various times ; that is to say, we might suppose particles 
to differ only in position, not in quality. At first, this was hardly 
more than a definition of the word “physics”; when it was 
necessary to take account of qualitative differences, we were 
concerned with a different subject, called “chemistry”. During 
the present century, however, the modern theory of the atom 
has reduced chemistry, theoretically, to physics. This has enor- 



mously extended the scope of the hypothesis that different 
particles of matter differ only in position. 

Does this hypothesis apply also to physiology, or is the be- 
haviour of living matter subject to laws different from those 
governing dead matter? Vitalists maintain the latter view, but I 
think the former has the greater weight of authority in its favour. 
What can be said is that, wherever a physiological process is 
understood, it is found to follow the laws of physics and chemistry, 
and that, further, there is no physiological process which is clearly 
not explicable by these laws. It is therefore the best hypothesis 
that physiology is reducible to physics and chemistry. But this 
hypothesis has not nearly the same degree of certainty as the 
reduction of chemistry to physics. 

I shall assume henceforth that the first of the seventeenth- 
century working hypotheses, which may be called the hypothesis 
of the homogeneity of matter, applies throughout the physical 
world, and to living as well as dead matter. I shall not constantly 
repeat that this theory is not certainly true ; this is to be taken as 
said once for all. I assume the theory because the weight of 
evidence, though not conclusive, seems to me strongly in its 

The second of the working hypotheses of the seventeenth 
century may be called the hypothesis of the independence of 
causes; it is embodied in the parallelogram law. In its simplest 
form it says such things as: If you walk for a minute on the deck 
of a moving ship, you will reach the same point, relatively to the 
water, as you would if first you stood still for a minute while the 
ship moved, and then the ship stood still for a minute while you 
did your walk on the deck. More generally, when a body is subject 
to several forces, the result of their all acting at once for a given 
length of time is the same as would be the result of their all 
acting by turns, each for the given length of time — or rather, if 
the given length of time is very short this will be nearly true, and 
the shorter the time the more nearly true it will become. For 
instance, the moon is attracted both by the earth and by the sun ; 
in one second, it will move very nearly as if, for one second, it 
were not attracted by either, but went on moving as before, then 
for another second it were to move as if (starting from rest) it 
were attracted by the earth only, then for another second as if 
(starting from rest) it were attracted by the sun only. If we take 

2I 5 

human knowledge: its scope and limits 

a shorter time than a second this will be more nearly true, 
approaching the limit of complete truth as the period of time is 
indefinitely diminished. 

This principle is of the utmost importance technically. It 
enables us, when we have studied the effects of a number of 
separate forces each acting singly, to calculate the effect of their 
all acting together. It is the basis of the mathematical methods 
employed in traditional physics. But it must be said that it is not 
self-evident, except in simple cases like that of the man walking 
on the deck of the ship. In other cases, it is to be believed if it 
works, but we ought not to be surprised if we find that it some- 
times does not work. In the quantum theory of the atom it has had 
to be abandoned, though this is perhaps not definitive. However 
that may be, this second working hypothesis is much less securely 
established than the first. It holds, at least approximately, over 
a wide field, but there is no good ground for believing that it holds 

The present century has somewhat modified the assumptions of 
physics. First, there is a four-dimensional manifold of events, 
instead of the two manifolds of space and time; second, causal 
laws do not suffice to determine individual events, but only 
statistical distributions; third, change is probably discontinuous. 
These modifications would be more important to us than they are, 
but for the fact that the second and third only apply effectively 
to microscopic phenomena, while the physical occurrences, such 
as speaking, which are associated with “mental” events, are 
macroscopic. Therefore if a human body works wholly in accor- 
dance with physical laws, it will still be correct to use the laws of 
classical physics to determine what a man will say, and generally 
what will be the large-scale motions of his body. 

This brings us to the problem of the relation of mind and 
matter, since perception is commonly considered “mental” while 
the object perceived and the stimulus to perceiving are considered 
“physical”. My own belief is that there is no difficulty whatever 
about this problem. The supposed difficulties have their origin 
in bad metaphysics and bad ethics. Mind and matter, we are told, 
are two substances, and are utterly disparate. Mind is noble, 
matter is base. Sin consists in subjection of the mind to the body. 
Knowledge, being one of the noblest of mental activities, cannot 
depend upon sense, for sense marks a form of subjection to 



matter, and is therefore bad. Hence the Platonic objection to 
identifying knowledge with perception. All this, you may think, 
is antiquated, but it has left a trail of prejudices hard to overcome. 

Nevertheless, the distinction of mind and matter would hardly 
have arisen if it had not some foundation. We must seek, therefore, 
for one or more distinctions more or less analogous to the distinc- 
tion between mind and matter. I should define a “mental” 
occurrence as one which can be known without inference. But let 
us examine some more conventional definitions. 

We cannot use the Cartesian distinction between thought and 
extension, if only on Leibniz’s ground, that extension involves 
plurality and therefore cannot be an attribute of a single substance. 
But we might try a somewhat analogous distinction. Material 
things, we may say, have spatial relations, while mental things 
do not. The brain is in the head, but thoughts are not — so at 
least philosophers assure us. This point of view is due to a con- 
fusion between different meanings of the word “space”. Among 
the things that I see at a given moment there are spatial relations 
which are a part of my percepts; if percepts are “mental”, as I 
should contend, then spatial relations which are ingredients of 
percepts are also “mental”. Naive realism identifies my percepts 
with physical things; it assumes that the sun of the astronomers 
is what I see. This involves identifying the spatial relations of my 
percepts with those of physical things. Many people retain this 
aspect of naive realism although they have rejected all the rest. 

But this identification is indefensible. The spatial relations 
of physics hold between electrons, protons, neutrons, etc., which 
we do not perceive; the spatial relations of visual percepts hold 
between things that we do perceive, and in the last analysis be- 
tween coloured patches. There is a rough correlation between 
physical space and visual space, but it is very rough. First: depths 
become indistinguishable when they are great. Second: the 
timing is different; the place where the sun seems to be now 
corresponds to the place where the physical sun was eight minutes 
ago. Third : the percept is subject to changes which the physicist 
does not attribute to changes in the object, e.g. those brought 
about by clouds, telescopes, squinting, or closing the eyes. The 
correspondence between the percept and the physical 6bject is 
therefore only approximate, and it is no more exact as regards 
spatial relations than it is in other respects. The sun of the 


HUMAN knowledge: its scope and limits 

physicist is not identical with the sun of my percepts, and the 
93,000,000 miles that separate it from the moon are not identical 
with the spatial relation between the visual sun and the visual 
moon when I happen to see both at once. 

When I say that something is “outside” me, there are two 
different things that I may mean. I may mean that I have a 
percept which is outside the percept of my body in perceptual 
space, or I may mean that there is a physical object which is 
outside my body as a physical object in the space of physics. 
Generally there is a rough correspondence between these two. 
The table that I see is outside my body as I see it in perceptual 
space, and the physical table is outside my physical body in 
physical space. But sometimes the correspondence fails. I dream, 
say, of a railway accident: I see the train falling down an em- 
bankment, and I hear the shrieks of the injured. These dream- 
objects are genuinely and truly “outside” my dream body in my 
own perceptual space. But when I wake up I find that the whole 
dream was due to a noise in my ear. And when I say that the noise 
is in my ear, I mean that the physical source of the sound that I 
experience is “in” my ear as a physical object in physical space. 
In another sense, we might say that all noises are in the ear, but 
if we confuse these two senses the result is an inextricable tangle. 

Generalizing, we may say that my percept of anything other 
than my body is “outside* * the percept of my body in perceptual 
space, and if the perception is not misleading the physical object 
is “outside** my physical body in physical space. It does not 
follow that my percept is outside my physical body. Indeed, such 
a hypothesis is prima facie meaningless, although, as we shall see. 
a meaning can be found for it, and it is then false. 

We can now begin to tackle our central question, namely, what 
do we mean by a “percept**, and how can it be a source of know- 
ledge as to something other than itself? 

What is a “percept**? As I use the word, it is what happens 
when, in common-sense terms, I see something or hear some- 
thing or otherwise believe myself to become aware of something 
through my senses. The sun, we believe, is always there, but I 
only sometimes see it: I do not see it at night, or in cloudy 
weather, or when I am otherwise occupied. But sometimes I see 
it. All the occasions on which I see the sun have a certain re- 
semblance to each other, which enabled me in infancy to learn 



to use the word “sun” on the right occasions. Some of the re- 
semblances between different occasions when I see the sun are 
obviously in me; for example, I must have my eyes open and turn 
in the right direction. These, therefore, we do not regard as 
properties of the sun. But there are other resemblances which, so 
far as common sense can discover, do not depend upon us ; when 
we see the sun, it is almost always round and bright and hot. The 
few occasions when it is not are easily explicable as due to fog or 
to an eclipse. Common sense therefore says: there is an object 
which is round and bright and hot; the kind of event called 
“seeing the sun” consists in a relation between me and this object, 
and when this relation occurs I am “perceiving” the object. 

But at this point physics intervenes in a very awkward way. It 
assures us that the sun is not “bright” in the sense in which we 
usually understand the word ; it is a source of light-rays which have 
a certain effect upon eyes and nerves and brains, but when this 
effect is absent because the light-rays do not encounter a living 
organism, there is nothing that can be properly called “brightness”. 
Exactly the same considerations apply to the words “hot” and 
“round” — at least if “round” is understood as a perceptible 
quality. Moreover, though you see the sun now, the physical 
object to be inferred from your seeing existed eight minutes ago; 
if, in the intervening minutes, the sun had gone out, you would 
still be seeing exactly what you are seeing. We cannot therefore 
identify the physical sun with what we see ; nevertheless what we 
see is our chief reason for believing in the physical sun. 

Assuming the truth of physics, what is there in its laws that 
justifies inferences from percepts to physical objects? Before we 
can adequately discuss this question, we must determine the 
place of percepts in the world of physics. There is here a peculi- 
arity : physics never mentions percepts except when it speaks of 
empirical verification of its laws ; but if its laws are not concerned 
with percepts, how can percepts verify them? This question 
should be borne in mind during the following discussions. 

The question of the position of percepts in the causal chains of 
physics is a different one from that of the cognitive status of 
percepts, though the two are interconnected. At the moment I am 
concerned with the location of percepts in causal chains. Now 
a percept — say hearing a noise — has a series of antecedents, which 
travel in space-time from the physical source of the noise through 


HUMAN knowledge: its scope and limits 

the air to the ears and brain. The experience which we call 
“hearing the noise ,, is as nearly as can be determined simultaneous 
with the cerebral term of the physical causal chain. If the noise is 
of the kind to call forth a bodily movement, the movement begins 
almost immediately after “hearing the noise”. If we are going to 
fit “hearing the noise” into a physical causal chain, we must 
therefore connect it with the same region of space-time as that of 
the accompanying cerebral events. And this applies also to the 
noise as something perceived. The only region of space-time with 
which this noise has any direct connection is the present state of 
the hearer’s brain ; the connection with the physical source of the 
sound is indirect. Exactly the same argument applies to things 

I am anxious to minimize the metaphysical assumptions to be 
made in this connection. You may hold that mind and matter 
interact, or that, as the Cartesians contended, they run in parallel 
series, or that, as materialists believe, mental occurrences are 
mere concomitants of certain physical occurrences, determined 
by them but having no reciprocal influence on physical events. 
What you hold in these respects has no bearing on the point that 
I am making. What I am saying is something which is obvious 
to educated common sense, namely that, whether we consider the 
percept or the simultaneous state of the brain, the causal location 
of either is intermediate between occurrences in afferent nerves 
constituting the stimulus, and occurrences in efferent nerves 
constituting the reaction. 

This applies not only to the perceiving, which we naturally 
regard as “mental”, but to what we experience when we perceive. 
That is to say, it applies not only to “seeing the sun”, but also 
to the sun, if we mean by “the sun” something that a human 
being can experience. The astronomer’s sun is inferred, it is not 
hot or bright, and it existed eight minutes before what is called 
seeing it. If I see the sun and it makes me blink, what I see is not 
93,000,000 miles and eight minutes away, but is causally (and 
therefore spatio-temporally) intermediate between the light-waves 
striking the eye and the consequent blinking. 

The dualistic view of perception, as a relation of a subject to 
an object, is one which, following the leadership of William 
James, empiricists have now for the most part abandoned. The 
distinction between “seeing the sun” as a mental event, and the 



immediate object of my seeing, is now generally rejected as 
invalid, and in this view I concur. But many of those who take 
the view that I take on this point nevertheless inconsistently 
adhere to some form of naive realism. If my seeing of the sun is 
identical with the sun that I see, then the sun that I see is not the 
astronomer’s sun. For exactly the same reasons, the tables and 
chairs that I see, if they are identical with my seeing of them, are 
not located where physics says they are, but where my seeing is. 
You may say that my seeing, being mental, is not in space; if 
you do, I will not argue the point. But I shall none the less insist 
that there is one, and only one, region of space-time with which 
my seeing is always causally bound up, and that is my brain at 
the time of the seeing. And exactly the same is true of all objects 
of sense-perception. 

We are now in a position to consider the relation between a 
physical occurrence and the subsequent occurrence popularly 
regarded as seeing it. Consider, say, a flash of lightning on a 
dark night. The flash, for the physicist, is an electrical discharge, 
which causes electromagnetic waves to travel outward from the 
region where it has taken place. These waves, if they meet no 
opaque matter, merely travel further and further; but when they 
meet opaque matter their energy undergoes transformations into 
new forms. When they happen to meet a human eye connected 
with a human brain, all sorts of complicated things happen, which 
can be studied by the physiologist. At the moment when this 
causal process reaches the brain, the person to whom the brain 
belongs “sees” the flash. This person, if he is unacquainted with 
physics, thinks that the flash is what takes place when he “sees” 
the flash ; or rather, he thinks that what takes place is a relation 
between himself and the flash, called “perceiving” the flash. If 
he is acquainted with physics, he does not think this, but he still 
holds that the sort of thing that takes place when he “sees” the 
flash gives an adequate basis for knowledge of the physical world. 

We can now at last tackle the question: How, and to what 
extent, can percepts be a source of knowledge as to physical 
objects? A percept, we have agreed, comes at the end of a causal 
chain which starts at the object. (Of course no causal chain really 
has either a beginning or an end. From another point of view the 
percept is a beginning; it begins the reaction to a stimulus.) If 
the percept is to be a source of knowledge of the object, it must 


HUMAN knowledge: its scope and limits 

be possible to infer the cause from the effect, or at least to infer 
some characteristics of the cause. In this backward inference 
from effect to cause, I shall for the present assume the laws of 

If percepts are to allow inferences to objects, the physical 
world must contain more or less separable causal chains. I can 
see at the present moment various things — sheets of paper, books, 
trees, walls, and clouds. If the separateness of these things in my 
visual field is to correspond to a physical separateness, each of 
them must start its own causal chain, arriving at my eye without 
much interference from the others. The theory of light assures 
us that this is the case. Light- waves emanating from a source will, 
in suitable circumstances, pursue their course practically un- 
affected by other light-waves in the same region. But when light- 
waves encounter a reflecting or refracting object this independence 
of the medium disappears. 

This is important in deciding what the object is that we are 
supposed to see. In the daytime, practically all the light that 
reaches the eye comes ultimately from the sun, but we do not say 
that we are seeing only the sun. We are seeing the last region 
after which the course of the light was virtually unimpeded until 
it reached the eye. When light is reflected or scattered, we con- 
sider, as a rule, that it makes us see the last object from which it 
is reflected or scattered ; when it is refracted, we consider that we 
are still seeing the previous source, though inaccurately. Re- 
flected light, however, is not always taken as giving perception 
of the reflector; it is not so taken when the reflection is accurate, 
as in a mirror. What I see when I shave I consider to be my own 
face. But when sunlight is reflected on an outdoor landscape it 
gives me much more information about the things in the landscape 
than about the sun, and I therefore consider that I am perceiving 
the things in the landscape. 

In a lesser degree similar things may be said about sound. We 
distinguish between hearing a sound and hearing an echo of it. 
If the sun were as chromatically noisy as it is bright, and if terres- 
trial things were resonant only to certain of its notes, we should 
say that we were hearing the things, not the sun, when they gave 
characteristic sound-reflections. 

The other senses do not give the same kind of perception of 
distant objects or of intermediate links in causal chains, because 



they are not concerned with physical processes having the peculiar 
kind of independence that is characteristic of wave motions. 

From what we have been saying it is clear that the relation of 
a percept to the physical object which is supposed to be perceived 
is vague, approximate, and somewhat indefinite. There is no 
precise sense in which we can be said to perceive physical objects. 

The question of perception as a source of knowledge can be 
merged in a wider question : How far, and in what circumstances, 
can one stage in a physical process be a basis for inferring an 
earlier stage ? Clearly this can only happen in so far as the process 
in question is independent of other processes. That processes can 
be thus independent is perhaps surprising. We see the separate 
stars because the light that starts from each travels on through 
regions full of other trails of light, and yet retains its independence. 
When this independence fails, we see a vague blur, like the milky 
way. In the case of the milky way, the independence does not fail 
till we reach the physiological stage; that is why telescopes can 
separate the different stars of the milky way. But the independence 
of the light from different parts of one star cannot be restored by 
telescopes ; that is why stars have no measurable apparent magni- 

Our perceptive apparatus, as studied by the physiologist, can 
to some extent be ignored by the physicist, because it can be 
treated as approximately constant. It is not of course really 
constant. By squinting I can see two suns, but I do not imagine 
that I have performed an astronomical miracle. If I close my eyes 
and turn my face to the sun, I see a vague red glare; this change 
in the sun's appearance I attribute to myself, not to the sun. 
Things look different when I see them out of the corner of my eye 
from what they do when I focus on them. They look different to 
short-sighted and to long-sighted people. And so on. But common 
sense learns to distinguish these subjective sources of variation 
in the percept from those that are due to variation in the physical 
object. Until we learn to draw, we think that a rectangular object 
always looks rectangular; and we are right, in the sense that an 
animal inference causes us to judge it to be rectangular. 

Science deals with these matters by assuming a normal observer 
who is to some extent a fiction, like the economic man, but not so 
completely a fiction as to be practically useless. When a normal 
observer sees a difference between two objects, for example that 


human knowledge: its scope and limits 

one looks yellow and the other looks blue, this difference is assumed 
to have its source in a difference in the objects, not in the subjective 
perceptive apparatus of the observer. If, in a given case, this 
assumption is erroneous, it is held that multiplicity of observations 
by a multitude of observers will correct it. By such methods, the 
physicist is enabled to treat our perceptive apparatus as the source 
of a constant error, which, because it is constant, is for many 
purposes negligible. 

The principles which justify the inference from percepts to 
physical objects have not been sufficiently studied. Why, for 
example, when a number of people see the sun, should we believe 
that there is a sun outside their percepts, and not merely that 
there are laws determining the circumstances in which we shall 
have the experience called “seeing the sun”? 

Here we come up against a principle which is used both by 
science and by common sense, to the effect that, when a number 
of phenomena in separated parts of space-time are obviously 
causally interconnected, there must be some continuous process 
in the intervening regions which links them all together. This 
principle of spatio-temporal continuity needs to be re-examined 
after we have considered the inference from perceptual to physical 
space. In the meantime, it can be accepted as at least a first step 
towards formalizing inference from perceptual to physical objects 

I will conclude with a summary of the present Chapter. 

Our main question was : If physics is true, how can it be known, 
and what, besides physics, must we know to infer physics? This 
problem arises through the physical causation of perception, 
which makes it probable that physical objects differ greatly from 
percepts; but if so, how can we infer physical objects from per- 
cepts? Moreover, since perceiving is considered to be “mental” 
while its causes are “physical”, we are confronted with the old 
problem of the relation between mind and matter. My own belief 
is that the “mental” and the “physical” are not so disparate as is 
generally thought. I should define a “mental” occurrence as one 
which some one knows otherwise than by inference ; the distinction 
between “mental” and “physical” therefore belongs to theory of 
knowledge, not to metaphysics. 

One of the difficulties which have led to confusion was failure 
to distinguish between perceptual and physical space. Perceptual 
space consists of perceptible relations between parts of percepts, 



whereas physical space consists of inferred relations between 
inferred physical things. What I see may be outside my percept 
of my body, but not outside my body as a physical thing. 

Percepts, considered causally, are between events in afferent 
nerves (stimulus) and events in efferent nerves (reaction); their 
location in causal chains is the same as that of certain events in 
the brain. Percepts as a source of knowledge of physical objects 
can only serve their purpose in so far as there are separable, more 
or less independent, causal chains in the physical world. This 
only happens approximately, and therefore the inference from 
percepts to physical objects cannot be precise. Science consists 
largely of devices for overcoming this initial lack of precision on 
the assumption that perception gives a first approximation to 
the truth. 


Chapter V 


T he purpose of this Chapter is to consider those features 
of crude experience which form the raw material of the 
concept of time, which has to go through a long elaboration 
before it is fit to appear in physics or history. There are two 
sources of our belief in time ; the first is the perception of change 
within one specious present, the other is memory. When you 
look at your watch, you can see the second-hand moving, but 
only memory tells you that the minute-hand and hour-hand 
have moved. Shakespeare’s timepieces had no second-hand, as 
appears from the lines : 

Ah! yet doth beauty, like a dial hand, 

Steal from his figure, and no pace perceiv’d. 

“Pace perceiv’d” is only possible when the motion is so rapid 
that, though the beginning and end are noticeably different, 
the lapse of time is so short that both are parts of one sensation. 
No sensation, not even that caused by a flash of lightning, is 
strictly instantaneous. Physiological disturbances die down 
gradually, and the length of time during which we see a flash 
of lightning is much greater than the length of time occupied 
by the physical phenomenon. 

The relation of “preceding”, or of “earlier-and-later”, is an 
element in the experience of perceiving a change, and also in 
the experience of remembering. Strictly speaking, we ought 
also to add immediate expectation, but this is of less importance. 
When I see a rapid movement, such as that of a falling star, or 
of cloud-shadows in a landscape, I am aware that one part of 
the movement is earlier than another, in spite of the whole being 
comprised within one specious present; if I were not aware of 
this, I should not know whether the movement had been from 
A to B or from B to A, or even that change had occurred. When 
a movement is sufficiently rapid we do not perceive change: 
if you spin a penny very efficiently, it takes on the appearance 
of a diaphanous sphere. If a motion is to be perceived, it must 
be neither too fast nor too slow. If it satisfies this condition, 



it provides experiences from which it is possible to obtain 
ostensive definitions of the words for temporal relations: “preced- 
ing”, “succeeding”, “before”, “after”, “earlier”, “later”. When 
these words have come to be understood, we can understand 
such sentences as “A precedes B” even when A and B are not 
part of one specious present, provided we know what is meant 
by “A” and what by “B”. 

But one specious present is a very small part of one man’s life, 
and for longer periods of time within our own experience we 
rely upon memory. In fact, of course, we rely upon a great deal 
besides memory. In regard to past engagements noted in my 
diary, I can infer their time-order and their distance from the 
present by the dates at which they are recorded. This, however, 
is a process presupposing considerable knowledge, whereas I 
am concerned at present with the data upon which our knowledge 
concerning time is based. Within limits, and with a considerable 
risk of error, we can place our memories in a time-order by the 
way they “feel”. Suppose we have just had a conversation, 
beginning amicably, but ending in a violent quarrel, and suppose 
the person with whom we were conversing has flounced out of 
the room in a rage. We can go over the whole conversation in 
retrospect, thinking “at this point I said the wrong thing”, or 
“at that point the remark he allowed himself was unpardonable”. 
Our memory, in fact, is not of a heap of events, but of a series , 
and often there can be no reasonable doubt that the time-order 
supplied by our memory is correct. 

There is here, however, a complication which has been too 
often overlooked. All my recollections occur nbw 9 not at the 
times when the recollected events occurred.^The time-order of 
the past events, in so far as I can know it by means of memory, 
must be connected with a quality of my recollections: some 
must feel recent and others must feel remote. It must be by 
means of this felt quality of recentness or remoteness that I 
place remembered events in a series when I am relying upon 
memory alone. In travelling from percepts towards “the dark 
backward and abysm of time”, the present contents of my mind 
have an order, which I believe to be correlated, roughly at any 
rate, with the objective time-order of the events to which my 
recollections refer. This order in the present contents of my 
mind, which, by means of expectation, may be extended into 


HUMAN knowledge: its scope and limits 

the future, may be called “subjective” time. Its relations to 
objective time are difficult, and demand discussion. 

St. Augustine, whose absorption in the sense of sin led him 
to excessive subjectivity, was content to substitute subjective 
time for the time of history and physics. Memory, perception, 
and expectation, according to him, made up all that there is of 
time. But obviously this won’t do. All his memories and all his 
expectations occurred at about the time of the fall of Rome, 
whereas mine occur at about the time of the fall of industrial 
civilization, which formed no part of the Bishop of Hippo’s 
expectations. Subjective time might suffice for a solipsist of the 
moment, but not for a man who believes in a real past and future, 
even if only his own. My momentary experience contains a 
space of perception, which is not the space of physics, and a 
time of perception and recollection, which is not the time of 
physics and history. My past, as it occurred, cannot be identified 
with my recollections of it, and my objective history, which was 
in objective time, differs from the subjective history of my present 
recollections, which, objectively, is all now. 

That memory is in the main veridical is, in my opinion, one 
of the premisses of knowledge. What this premiss asserts is, 
or implies, broadly speaking, that a present recollection is as a 
rule correlated with a past event. Obviously this is not logically 
necessary. I might have come into existence a few moments ago, 
complete with just those recollections which I then had. If the 
whole world came into existence then, just as it then was, there 
will never be anything to prove that it did not exist earlier; in 
fact, all the evidence that we now have in favour of its having 
existed earlier, we should then have. This illustrates what I mean 
by saying that memory is a premiss, for we are none of us 
prepared for a moment to entertain the supposition that the 
world began five minutes ago. We do not entertain the supposi- 
tion, because we are persuaded that, as a rule, when we recollect, 
something resembling our present recollection occurred at a 
time which is objectively past. 

I said a moment ago that the general trustworthiness of memory 
is a premiss of human knowledge. We might find at a later stage 
that it can be subsumed under a wider premiss, but for the present 
this possibility may be ignored. What does, however, need to 
be considered at this stage is the relation of confidence in 



particular memories to the postulate that memory is trustworthy 
as a rule, or in certain specified circumstances. 

When I remember something, I do not first take note of my 
present state of mind, then reflect that memory is usually 
veridical, and finally infer that something like what I am re- 
collecting occurred in the past. On the contrary, what happens 
when I remember is belief that something happened in the past. 
What I am concerned with in this Chapter is (a) analysis of such 
beliefs, and (b) statement of what is meant when such a belief is 
said to be true. Neither of these questions is as simple as seems 
to be generally supposed. 

Memories often float through the mind as mere images, un- 
accompanied by belief, but I am concerned only with memories 
that are believed. Let us take a concrete illustration. Suppose I 
have seen my child very nearly run over by a motor-car, but in 
fact unhurt, and suppose that in the following night I have a 
nightmare in which the child is killed. When I wake up, I think, 
with inexpressible relief: “ This did not occur; that occurred.” 

A good deal of clarification is necessary before we can arrive 
at the core of the problems raised by this illustration. To begin 
with, when we say “this did not occur”, we are not denying that 
the nightmare occurred; in so far as we are remembering the 
nightmare as a private experience, our memory is quite correct. 
The nightmare, however, did not have the context that waking 
experiences have: it had no context whatever in the life of the 
child, or of any person except myself and such persons as had to 
listen while I related it, and in my own life its context sharply 
ended when I woke, instead of being prolonged through years 
of sorrow. This is the sort of thing we mean by saying that the 
nightmare was only a dream. 

But all this is irrelevant to our problem of memory, and I 
have only mentioned it in order to make clear what is relevant 
and what is not. When I remember the nightmare, my memory 
is veridical ; I am only misled if I suppose the nightmare to have 
had the sort of context that a similar waking experience would 
have had. An error of memory occurs only when we believe that, 
in the past, we had some experience which in fact we did not 
have, and when, further, we believe this in the specific way that 
is called “remembering”, as opposed to the way that comes when 
we read records of forgotten events, or have to listen to aunts 


human knowledge: its scope and limits 

relating our exploits as children. Such errors of memory un- 
doubtedly occur. I will not insist upon George IV remembering 
that he was at the battle of Waterloo; coming nearer home, I 
know that when, too late, I have thought of a witty retort, I 
find a tendency to remember that I actually made it, which I 
only resist by a considerable moral effort. When two people 
independently report an acrimonious conversation, each will 
usually falsify the truth in a way favourable to his self-esteem. 
And even memories having little emotional interest can often 
be proved by records to be inaccurate. 

But the most convincing examples of false memories are 
supplied by dreams, though not by the nightmare which I 
supposed a moment ago. Let us alter the nightmare: I now do 
not dream that I see the child killed, but that, having seen this, 
it is my duty to tell the child’s mother what has happened. This 
is just as much a nightmare, but now the false belief in my dream 
is not merely as to the context of my experience, but as to my 
very own past experience. When I dream that I see the child 
run over, I do have the experience, though it does not have its 
usual concomitants; but when I dream that I saw the child 
run over, I never did have the experience that in my dream I 
am remembering. This is a genuine case of false memory, and 
shows that memory alone cannot make it certain that what is 
remembered really occurred, however much we may whittle down 
what is remembered to its core of purely personal experience. 

This example, I hope, will make it clear what I mean by “sub- 
jective” time, and what is the problem of its relation to objective 
time. In dreams, as in waking life, there is a difference between 
perceiving and remembering. The perceiving and the remembering 
do really occur in dreams, and so far as the perceiving is concerned 
we do not have to suppose that dreams deceive us as to our own 
experiences: what, in dreams, we see and hear, we do in fact 
see and hear, though, owing to the unusual context, what we see 
and hear gives rise to false beliefs. Similarly what we remember 
in dreams we do really remember, that is to say, the experience 
called “remembering” does occur. In the dream, this remembering 
has a quality differing from that of dream-perception, and in 
virtue of this quality the remembering is referred to the past. 
But the quality is not that of genuine pastness which belongs to 
the events of history; it is that of subjective pastness, in virtue 



of which the present remembering is judged (falsely) to refer to 
something that is objectively past. 

This quality of subjective pastness belongs to waking memories 
as well as to those of dreams, and is what makes them subjectively 
distinguishable from perceptions. It is a quality which is capable 
of degrees: our memories feel more remote or less remote, and 
can be arranged in a series by this qualitative difference. But since 
all our memories are now from the point of view of history, this 
subjective time-order is wholly distinct from the objective time- 
order, though we hope that there is a certain degree of correspond- 
ence between the two. 

I can perceive a remembering, but I cannot perceive what is 
remembered. Remembering consists of remembering “some- 
thing”. I want now to analyse remembering, and especially to 
consider this relation to “something”. In short: what do we 
mean when, thinking of some past occurrence, we judge “ that 
occurred”? What can “that” be? 

The difficulty is that, to know what we mean by “that occurred”, 
the word “that” must refer to some present content of the mind, 
whereas, if the word “occurred” is justified, the word “that” must 
refer to something in the past. Thus it would seem that the word 
“that” must refer to something which is both past and present. 
But we are in the habit of thinking that the past is dead, and that 
nothing past can also be present. What, then, do we mean when 
we judge “that occurred”? 

There are two different possible answers, connected with the 
two different theories as to proper names which we considered 
in an earlier chapter. If we consider that, in describing the struc- 
ture of the world, the terms which must be merely named must 
include “events”, which are uniquely defined by their spatio- 
temporal position, and are logically incapable of recurring, then 
we must say that the phrase “that occurred” is inaccurate, and 
should be replaced by “something very like that occurred”. 
If, on the other hand, we hold that an “event” can be defined 
as a bundle of qualities, each and all of which may recur, then 
“that occurred” may be completely accurate. If, for example, 
I see a rainbow on two occasions, and I see some shade of colour 
near the middle on one occasion, I probably saw the very same 
shade on the other occasion. If, then, remembering the earlier 
rainbow while I am seeing the later one, I say, of some shade of 


human knowledge: its scope and limits 

colour that I am now seeing, “that occurred on the previous 
occasion”, what I say may be exactly true. 

Either of these answers will solve the particular difficulty 
with which we have been immediately concerned, and for the 
present I shall not attempt to decide between them. They leave 
open the question what is meant by the word “occurred”; I 
shall deal with this when I come to consider public time. 

It should be observed that what we mean by “the past” in the 
historical sense is understood by us in virtue of the experience 
of succession within one specious present. It is this experience 
that makes us understand the word “precedes”. We can then 
understand: “if y is in the specious present, x precedes y”. We 
can therefore understand what is meant by saying that x precedes 
everything in the specious present, i.e. that x is in the past. 
The essential point is that the time that occurs in the specious 
present is objective, not subjective. 

We can now sum up the discussion of this Chapter. 

There are two sources for our knowledge of time. One is the 
perception of succession within one specious present, the other 
is memory. Remembering can be perceived, and is found to have 
a quality of greater or less remoteness, in virtue of which all my 
present memories can be placed in a time-order. But this time is 
subjective, and must be distinguished from historical time. 
Historical time has to the present the relation of “preceding”, 
which I know from the experience of change within one specious 
present. In historical time, all my present memories are now , 
but in so far as they are veridical they point to occurrences in 
the historical past. There is no logical reason why any memories 
should be veridical; so far as logic can show, all my present 
memories might be just what they are if there had never been 
any historical past. Our knowledge of the past therefore depends 
upon some postulate which is not to be discovered by mere 
analysis of our present rememberings. 


Chapter VI 


P sychology is concerned with space, not as a system of 
relations among material objects, but as a feature of our 
perceptions. If we could accept naive realism, this dis- 
tinction would have little importance : we should perceive material 
objects and their spatial relations, and the space that characterizes 
our perceptions would be identical with the space of physics. 
But in fact naive realism cannot be accepted, percepts are not 
identical with material objects, and the relation of perceptual 
to physical space is not identity. What the relation is, I shall 
consider presently; I am concerned, to begin with, only with 
space as it appears in psychology, ignoring all questions of 

It is clear that experience is what has led us to believe in the 
existence of spatial relations. Psychology is concerned to examine 
what experiences are relevant, and by what process of inference 
or construction we pass from such experiences to the space of 
common sense. Since a great part of the process occurs in early 
infancy, and is no longer remembered in later years, it is a some- 
what difficult matter of observation and inference to discover the 
character of the original experiences which give rise to the habits 
that adult common sense takes for granted. To take only the most 
obvious instance : we place things touched and things seen in one 
space, automatically and without reflection, but babies under the 
age of about three months seem unable to do so. That is to say, 
they do not know how to touch an object that they see and that 
is within their reach. It is only through frequent accidental 
contacts that they gradually learn the movements necessary to 
produce a tactile sensation when a visual sensation is given. 
Chickens, on the other hand, can do this from birth. 

We have to separate the crude material of sensation from the 
supplementation that it has acquired through experience and 
habit. When you see (let us say) an orange, you do not have 
merely a visual experience, but also expectations of touch, smell, 
and taste. You would have a violent shock of surprise if you 
found that it felt like putty, or smelt like a bad egg, or tasted 


HUMAN knowledge: its scope and limits 

like a beef-steak. You would be even more surprised if, like 
Macbeth’s dagger, it proved incapable of being touched. Such 
surprises show that expectations of non-visual sensations are 
part of what spontaneously happens to you when you have a 
visual sensation of a familiar kind. In the chicken, apparently, 
such expectations are in part due to its innate constitution. In 
human beings this happens much less, if at all; our expectations 
seem to be generated, mainly if not wholly, by experience. A 
visual sensation has at first, it would seem, a certain purity, and 
only gradually, through frequent collocations, acquires the 
penumbra of expectations connected with other senses that it 
has in adult life. And the same is true of other senses. 

It follows that the unitary space of common sense is a con- 
struction, though not a deliberate one. It is part of the business 
of psychology to make us aware of the steps in this construction. 

When we examine our momentary visual field, stripping it, 
as far as we can, of all the adjuncts derived from experience, we 
find that it is a complex whole in which the parts are inter- 
related in various ways. There are relations of right and left, 
up and down; there are also relations which we learn to interpret 
as depth. These relations all belong to the sensational datum. 
The best way to become aware of the sensational element in 
visual perceptions of depth is by the use of the stereoscope. 
When you look at the two separate photographs that are going 
to be seen together in the stereoscope, they both look flat, as 
they are; but when you see the combination in the stereoscope, 
you get the impression that things “stand out”, and that some 
are nearer than others. As a judgment this would of course be 
mistaken; the photographs are just as flat as they were before. 
But it is a genuine quality of the visual datum, and very instruc- 
tive as a help in showing how we arrive visually at estimates of 

By means of the three relations right-and-left, up-and-down, 
seeming-far-and-seeming-near, your momentary visual field can 
be arranged in a three-dimensional manifold. But far-and-near, 
estimated visually, is not capable of distinguishing except when 
one of the distances is very short; we cannot “see” that the sun 
is further off than the moon, or even than clouds which are not 
obscuring it. 

Other senses than sight supply other elements which contri- 



bute to the common-sense construction of space. When a part of 
the body is touched, we can tell, within limits, what part it is, 
without needing to look. (On the tongue or the finger-tips we 
can tell pretty accurately, on the back only vaguely.) This implies 
that touches in one part have a quality not belonging to touches 
in another part, and that the qualities appropriate to different 
parts have relations enabling us to arrange them in a two-dimen- 
sional order. Experience teaches us to connect sensations of touch 
with the visual sensations of seeing different parts of the body. 

Not only static sensations, such as we have mentioned, but 
also sensations of movement are involved in constructing 
common-sense space. Sensations of movement are of two sorts, 
active and passive — active when we have a feeling of muscular 
exertion, passive when the observed change seems independent 
of ourselves. When we move a part of our body and see it move, 
we have active and passive sensations at the same time. What 
I am calling passive sensations are only relatively passive; there 
is always the activity of attention, involving adjustment of the 
sense organs, except in the case of rather violent sensations. 
When you unexpectedly hit your head against a low doorway 
you are almost wholly passive, whereas when you listen carefully 
to a very faint sound the element of activity is considerable. 
(I am speaking of activity and passivity as elements in sensations, 
and am not inquiring into their causal status.) 

Movement is essential in enlarging our conception of space 
beyond our own immediate neighbourhood. The distance from 
where we are to some place may be estimated as an hour’s 
walk, three hours by train, or twelve hours by aeroplane. All 
such estimates assume fixed places. You can tell how long it 
takes from London to Edinburgh, because both retain fixed 
positions on the earth’s surface, but you cannot tell how long it 
will take to reach Mr. Jones, because he may move while you 
are on the way. All distances above a rather small minimum 
depend upon the assumption of immobility; it is partly the 
fact that this assumption is never quite true that necessitated 
the special theory of relativity, in which distance is between 
events, not bodies, and is a space-time distance, not a purely 
spatial distance. But such considerations take us beyond the 
scope of common sense. 

It is to be observed that the spatial relations given in sensation 


HUMAN knowledge: its scope and limits 

are always between data of the same sense. There is a spatial 
relation between two parts of the same visual field, or between 
two simultaneous pin-pricks on different parts of the hand; 
such spatial relations are within the realm of sensation, and are 
not learnt by experience. But between the tactual sensation of 
a pin-prick and the visual sensation of seeing the pin there is 
no direct sensational spatial relation, but only a correlation 
which human beings learn by experience. When you both see and 
feel a pin touching your hand it is only experience that enables 
you to identify the point of contact seen and the point of contact 
felt by touch. To say that they are the same place is convenient, 
but in psychology it is not strictly accurate: what is accurate 
is that they are correlated places in two different spaces, visual 
and tactual. It is true that in physical space only one place is 
involved, but this place lies outside our direct experience, and 
is neither visual nor tactual. 

The construction of one space in which all our perceptual 
experiences are located is a triumph of pre-scientific common 
sense. Its merit lies in its convenience, not in any ultimate truth 
that it may be supposed to possess. Common sense, in attributing 
to it a degree of non-conventional truth beyond what it actually 
has a right to claim, is in error, and this error, uncorrected, adds 
greatly to the difficulty of a sound philosophy of space. 

An even more serious error, committed not only by common 
sense but by many philosophers, consists in supposing that the 
space in which perceptual experiences are located can be identified 
with the inferred space of physics, which is inhabited mainly 
by things which cannot be perceived. The coloured surface 
that I see when I look at a table has a spatial position in the 
space of my visual field ; it exists only where eyes and nerves and 
brain exist to cause the energy of photons to undergo certain 
transformations. (The “where” in this sentence is a “where” 
in physical space.) The table as a physical object, consisting of 
electrons, positrons, and neutrons, lies outside my experience, 
and if there is a space which contains both it and my perceptual 
space, then in that space the physical table must be wholly 
external to my perceptual space. This conclusion is inevitable 
if we accept the view as to the physical causation of sensations 
which is forced on us by physiology and which we considered 
in an earlier chapter. 



The conception of one unitary space, Kant’s “infinite given 
whole”, is one which must be abandoned. The crude material 
available for empirical constructions contains several kinds of 
relations — more especially those between parts of one visual 
field or parts of one tactual field — each of which arranges its 
field in a manifold having the properties that pure mathematicians 
need for a geometry. By means of correlations — more especially 
between the visual and tactual place of an object which I simultan- 
eously see and touch — the various spaces generated by relations 
of parts of single sensational fields can be amalgamated into one 
space. To the making of this space experience of correlations is 
necessary; the kinds of relations given in single experiences no 
longer suffice. 

The common- sense world results from a further correlation, 
combined with an illegitimate identification. There is a correlation 
between the spatial relations of unperceived physical objects 
and the spatial relations of visual or other sensational data, and 
there is an identification of such data with certain physical objects. 
For example : I am sitting in a room, and I see — or at least common 
sense thinks I see — spatial relations between the pieces of furniture 
that it contains. I know that on the other side of the door there 
is a hall and a staircase. I believe that the spatial relations of 
things beyond the door — e.g. the relation “to-the-left-of” — 
are the same as those between the bits of furniture that I see; 
and further, I identify what I see with physical objects which 
can exist unseen, so that, if I am content with common sense, 
there is no gulf between the visual furniture and the unseen 
hall beyond the door. The two accordingly are thought to fit 
into one space, of which part is perceived while the rest is 

But in fact, if physics and physiology are to be believed, I 
do not “see” the furniture in my room except in a Pickwickian 
sense. When I am said to “see” a table, what really happens is 
that I have a complex sensation which is, in certain respects, 
similar in structure to the physical table. The physical table, 
consisting of electrons, positrons, and neutrons, is inferred, 
and so is the space in which it is located. It has long been a 
commonplace in philosophy that the physical table does not 
have the qualities of the sensational table: it has no colour, it 
is not warm or cold in the sense in which we know warmth 


HUMAN knowledge: its scope and limits 

and cold by experience, it is not hard or soft if “hard” and “soft” 
mean qualities given in tactile sensations, and so on. All this, 
I say, has long been a commonplace, but it has a consequence 
that has not been adequately recognized, namely that the space 
in which the physical table is located must also be different 
from the space that we know by experience. 

We say that the table is “outside” me, in a sense in which my 
own body is not. But in saying this we have to guard against an 
ambiguity due to the necessity of distinguishing between physical 
and psychological space. The visual table is “outside” my body 
in visual space, if “my body” is interpreted as what I see, and 
not as what physics takes to be my body. The physical table 
is “outside” my body if my body is interpreted as in physics, 
but has no direct or obvious spatial relation to my body as a visual 
object that I experience. When we come to consider the hall on 
the other side of the door, which I cannot see, we are wholly 
confined to the physical sense: the hall is outside my physical 
body in physical space, but is not, in any obvious sense, outside 
my sensational body in psychological space, because there is no 
sensational hall, and therefore the hall has no location whatever 
in psychological space. Thus while there are two senses in which 
the table is “outside” me, there is only one sense in which the 
hall is “outside” me. 

There is a further source of confusion, which is due to the fact 
that there are two quite divergent ways of correlating psychological 
and physical space. The obvious way is to correlate the place of 
the sensational table in psychological space with the place of the 
physical table in physical space, and for most purposes this is 
the more important correlation. But there is a quite different 
relation between the two kinds of space, and this other relation 
must be understood if confusions are to be avoided. Physical 
space is wholly inferential, and is constructed by means of causal 
laws. Physics starts with a manifold of events, some of which 
can be collected into series by physical laws; for example, the 
successive events constituting the arrival of a light-ray at suc- 
cessive places are bound together by the laws of the propagation 
of light. In such cases we use the denial of action at a distance, 
not as a physical principle, but as a means of defining space-time 
order. That is to say, if two events are connected by a causal 
law, so that one is an effect of the other, any third event which 



is a cause of the one and an effect of the other is to be placed 
between the two in space-time order. 

Consider now a single causal sequence, beginning with an 
external stimulus, say to the eye, continuing along afferent nerves 
to the brain, producing first a sensation and then a volition, 
followed by a current along efferent nerves and finally a muscular 
movement. This whole series, considered as one causal sequence, 
must, in physical space-time, occupy a continuous series of 
positions, and since the physiological terms of the series end and 
begin in the brain, the “mental” terms must begin and end in 
the brain. That is to say, considered as part of the manifold of 
events ordered in space-time by causal relations, sensations and 
volitions must be located in the brain. A point in space-time, 
following the theory to be developed in a subsequent chapter, 
is a class of events, and there is no reason why some of these 
events should not be “mental”. Our feeling to the contrary is 
only due to obstinate adherence to the mind-matter dualism. 

We can now sum up the above discussion. When I have the 
experience called “seeing a table”, the visual table has, primarily, 
a position in the space of my momentary visual field. Then, 
by means of experienced correlations, it has a position in a space 
which embraces all my perceptual experiences. Next, by means 
of physical laws it is correlated with a place in physical space- 
time, namely the place occupied by the physical table. Finally, 
by means of psychophysical laws it is related to another place 
in physical space-time, namely the place occupied by my brain 
as a physical object. If the philosophy of space is to avoid hopeless 
confusions, these different correlations must be kept carefully 

In conclusion, it should be observed that the twofold space 
in which percepts are located is closely analogous to the twofold 
time of memories. In subjective time, memories are in the past; 
in objective time, they are now. Similarly, in subjective space 
my percept of a table is over there, but in physical space it is here. 


Chapter VII 


C ommon sense believes that we know something about 
mind, and something about matter; it holds, further, 
that what we know of both is enough to show that they 
are quite different kinds of things. I hold, on the contrary, that 
whatever we know without inference is mental, and that the 
physical world is only known as regards certain abstract features 
of its space-time structure — features which, because of their 
abstractness, do not suffice to show whether the physical world 
is, or is not, different in intrinsic character from the world of 

I will begin with an attempt to state the common-sense point 
of view as clearly as is possible in view of the confusions that 
are essential to it. 

Mind — so common sense might say — is exhibited by persons 
who do and suffer various things. Cognitively, they perceive, 
remember, imagine, abstract, and infer; on the side of the 
emotions, they have feelings that are pleasurable and feelings 
that are painful, and they have sentiments, passions, and desires; 
volitionally, they can will to do something or will to abstain 
from doing something. All these occurrences can be perceived 
by the person to whom they happen, and all are to be classified 
together as “mental” events. Every mental event happens “to” 
some person, and is an event in his life. 

But in addition to perceiving “thoughts” — so common sense 
holds — we also perceive “things”, and events which are outside 
ourselves. We see and touch physical objects; we hear sounds 
which are also heard by other people, and therefore are not 
in us; when we smell a bad drain, other people do so too, unless 
they are plumbers. What we perceive, when it is outside our- 
selves, is called “physical”; this term includes both “things” 
which are “matter”, and events, such as a noise or a flash of 

Common sense also allows inferences to what is not perceived, 
at any rate by us, e.g. the centre of the earth, the other side of 
the moon, the thoughts of our friends, and the mental events 



that have produced historical records. An inferred mental event 
can be known without inference by the person to whom it happens. 
An inferred physical thing or event may or may not have been 
perceived by some one; some physical things, such as the centre 
of the earth, are held to have been never perceived. 

This common-sense view, while on the whole acceptable 
as regards mental events, requires radical alteration where 
physical events are concerned. What I know without inference 
when I have the experience called “seeing the sun” is not the 
sun, but a mental event in me. I am not immediately aware of 
tables and chairs, but only of certain effects that they have on me. 
The objects of perception which I take to be “external” to me, 
such as coloured surfaces that I see, are only “external” in my 
private space, which ceases to exist when I die — indeed my 
private visual space ceases to exist whenever I am in the dark 
or shut my eyes. And they are not “external” to “me”, if “me” 
means the sum-total of my mental events; on the contrary, they 
are among the mental events that constitute me. They are only 
“external” to certain other percepts of mine, namely those which 
common sense regards as percepts of my body; and even to 
these they are “external” only for psychology, not for physics, 
since the space in which they are located is the private space of 

In considering what common sense regards as perception of 
external objects, there are two opposite questions to be considered : 
first, why must the datum be regarded as private; second, what 
reason is there to take the datum as a sign of something which 
has an existence not dependent upon me and my perceptive 
apparatus ? 

The reasons for regarding the datum — say in sight or touch — 
as private, are twofold. On the one hand there is physics, which, 
starting with the intention of doing its best for naive realism, 
arrives at a theory of what goes on in the physical world which 
shows that there is no ground for supposing the physical table 
or chair to resemble the percept except in certain abstract 
structural respects. On the other hand there is the comparison 
of what different people experience when, according to common 
sense, they perceive the same thing. If we confine ourselves to 
the sense of sight, when two people are said to see the same 
table there are differences of perspective, differences of apparent 

241 Q 

HUMAN knowledge: its scope and limits 

size, differences in th way light is reflected, and so on. Thus 
at most the projective properties of the table are the same for a 
number of percipients, and even these are not quite the same 
if there is a refracting medium such as a steaming kettle, or our 
old friend the water that makes a stick look bent. If we consider, 
as common sense does, that the “same” object can be perceived 
by both sight and touch, the object, if it is to be really the same, 
must be still further removed from the datum, for a complex 
sight-datum and a complex touch-datum differ in intrinsic 
quality, and cannot be similar except in structure. 

Our second question is more difficult. If the datum in my 
perceptions is always private to me, why do I nevertheless regard 
it as a sign by means of which I can infer a physical “thing” or 
event which is a cause of my percept if my body is suitably 
placed, but does not form part of my immediate experience 
except partially in exceptional cases ? 

When we begin to reflect, we find ourselves with an unshake- 
able conviction that some of our sensations have causes external 
to our own body. Headache, toothache, and stomachache, we 
are willing to admit, have internal causes, but when we stub our 
toe or run into a post in the dark or see a flash of lightning we 
cannot easily make ourselves doubt that our experience has an 
external source. It is true that we sometimes come to think 
that this belief was mistaken, for instance if it occurs in dreams, 
or when we have a buzzing in the ears which sounds like the 
hum of telegraph wires. But such cases are exceptional, and 
common sense has discovered ways of dealing with them. 

What chiefly confirms us in our belief that most sensations 
have physical causes is, on the one hand, the quasi-publicity 
of many sensations, and, on the other hand, the fact that, if 
regarded as arising spontaneously, they seem completely erratic 
and unaccountable. 

As regards quasi-publicity, the argument is the opposite of 
that which proves the privacy of data : although two neighbouring 
men do not have exactly the same visual data, they have data 
which are very similar, and although visual and tactual qualities 
are different, the structural properties of an object seen are 
approximately identical with those of the same object touched. 
If you have models of the regular solids, the one which you can 
see to be a dodecahedron will be correctly named by an educated 



blind man after feeling it. Apart from the publicity concerned 
with different percipients, there is also what may be called 
temporal publicity in one man’s experience. I know that by taking 
suitable measures I can see St. Paul’s at any time; I know that 
the sun and moon and stars recur in my visual world, and so 
do my friends and my house and my furniture. I know that the 
differences between the times when I see these objects and the 
times when I do not are easily explicable as due to differences 
in me or my environment which do not imply any change in 
the objects. Such considerations confirm the common-sense 
belief that there are, in addition to mental events, things which 
are sources of similar percepts in different observers at one time, 
and often of the same observer at different times. 

As regards the irregularity of a world consisting only of data, 
this is an argument to which it is difficult to give precision. 
Roughly speaking, many sensations occur without any fixed 
antecedents in our own experience, and in a manner which 
suggests irresistibly that, if they have causes, these causes lie 
partly outside our experience. If you are hit on the head by a 
tile falling off a roof while you are walking below, you will 
experience a sudden violent pain which cannot be explained 
causally by anything of which you were aware before the accident 
happened. It is true that there are some extreme psycho-analysts 
who maintain that accidents only happen to people who have 
grown tired of life through reflecting on their sins, but I do not 
think such a view has many adherents. Consider the inhabitants 
of Hiroshima when the bomb burst: it cannot be that they had 
all reached a point in their psychological development which 
demanded disaster as the next step. To explain such an occur- 
rence causally, we must admit purely physical causes; if they 
are rejected, we must acquiesce in causal chaos. 

Such arguments may be reinforced by the considerations 
set forth above in the chapter on solipsism, showing that we 
must choose between two alternatives: either (a) no inferences 
from data to other events are to be admitted as valid, in which 
case we know far less than most solipsists suppose, and in fact 
a great deal less than we can force ourselves to regard as the 
minimum of our knowledge, or ( b ) there are principles of inference 
which allow us to infer things outside our own experience. 

Belief in the physical causation of sensations is also reinforced 


human knowledge: its scope and limits 

by the fact that, if this belief is rejected, there remains no reason 
for the acceptance of science in its broad outlines, while the 
refusal of such acceptance does not seem rational. 

Such are the broad considerations which lead us to look for a 
way of systematizing and rationalizing our common-sense pro- 
pensity to infer physical causes of sensations. 

The inferences from experiences to the physical world can, 
I think, all be justified by the assumption that there are causal 
chains, each member of which is a complex structure ordered 
by the spatio-temporal relation of compresence (or of conti- 
guity); that all the members of such a chain are similar in struc- 
ture ; that each member is connected with each other by a series 
of contiguous structures; and that, when a number of such 
similar structures are found to be grouped about a centre earlier 
in time than any of them, it is probable that they all have their 
causal origin in a complex event which is at that centre and has 
a structure similar to the structure of the observed events. I shall, 
at a later stage, endeavour to give greater precision to this assump- 
tion, and to show reasons for accepting it. For the present, to 
avoid verbiage, I shall treat it as though it were unquestionably 
correct, and on this basis I shall return to the relations between 
mental and physical events. 

When, on a common-sense basis, people talk of the gulf between 
mind and matter, what they really have in mind is the gulf between 
a visual or tactual percept and a “thought” — e.g. a memory, a 
pleasure, or a volition. But this, as we have seen, is a division 
within the mental world ; the percept is as mental as the “thought”. 
Slightly more sophisticated people may think of matter as the 
unknown cause of sensation, the “thing-in-itself” which certainly 
does not have the secondary qualities and perhaps does not have 
the primary qualities either. But however much they may 
emphasize the unknown character of the thing-in-itself, they 
still suppose themselves to know enough of it to be sure that it 
is very different from a mind. This comes, I think, of not having 
rid their imaginations of the conception of material things as 
something hard that you can bump into. You can bump into 
your friend's body, but not into his mind; therefore his body is 
different from his mind. This sort of argument persists imagina- 
tively in many people who have rejected it intellectually. 

Then, again, there is the argument about brain and mind, 



When a physiologist examines a brain, he does not see thoughts, 
therefore the brain is one thing and the mind which thinks is 
another. The fallacy in this argument consists in supposing 
that a man can see matter. Not even the ablest physiologist 
can perform this feat. His percept when he looks at a brain is 
an event in his own mind, and has only a causal connection 
with the brain that he fancies he is seeing. When, in a powerful 
telescope, he sees a tiny luminous dot, and interprets it as a vast 
nebula existing a million years ago, he realizes that what he 
sees is different from what he infers. The difference from the 
case of a brain looked at through a microscope is only one of 
degree: there is exactly the same need of inference, by means 
of the laws of physics, from the visual datum to its physical 
cause. And just as no one supposes that the nebula has any close 
resemblance to a luminous dot, so no one should suppose that 
the brain has any close resemblance to what the physiologist 

What, then, do we know about the physical world? Let us 
first define more exactly what we mean by a “physical” event. 
I should define it as an event which, if known to occur, is inferred, 
and which is not known to be mental. And I define a “mental” 
event (to repeat) as one with which some one is acquainted 
otherwise than by inference. Thus a “physical” event is one 
which is either totally unknown, or, if known at all, is not known 
to any one except by inference — or, perhaps we should say, is 
not known to be known to any one except by inference. 

If physical events are to suffice as a basis for physics, and, 
indeed, if we are to have any reason for believing in them, they 
must not be totally unknown, like Kant’s things-in-themselves. 
In fact, on the principle which we are assuming, they are known, 
though perhaps incompletely, so far as their space-time structure 
is concerned, for this must be similar to the space-time structure 
of their effects upon percipients. E.g. from the fact that the 
sun looks round in perceptual space we have a right to infer 
that it is round in physical space. We have no right to make a 
similar inference as regards brightness, because brightness is 
not a structural property. 

We cannot, however, infer that the sun is not bright— meaning 
by “brightness” the quality that we know in perception. The 
only legitimate inferences as regards the physical sun are 


HUMAN knowledge: its scope and limits 

structural; concerning a property which is not structural, such 
as brightness, we must remain completely agnostic. We may 
perhaps say that it is unlikely that the physical sun is bright, 
since we have no knowledge of the qualities of things that are 
not percepts, and therefore there seems to be an illimitable field 
of choice of possible qualities. But such an argument is so specula- 
tive that perhaps we ought not to attach much weight to it. 

This brings us to the question: Is there any reason, and if so 
what, for supposing that physical events differ in quality from 
mental events? 

Here we must, to begin with, distinguish events in a living 
brain from events elsewhere. I will begin with events in a living 

I assume, for reasons which will be given in Part IV, that a 
small region of space- time is a collection of compresent events, 
and that space-time regions are ordered by means of causal 
relations. The former assumption has the consequence that there 
is no reason why thoughts should not be among the events of 
which the brain consists, and the latter assumption leads to the 
conclusion that, in physical space, thoughts are in the brain. 
Or, more exactly, each region of the brain is a class of events, 
and among the events constituting a region thoughts are included. 
It is to be observed that, if we say thoughts are in the brain, we 
are using an ellipsis. The correct statement is that thoughts are 
among the events which, as a class, constitute a region in the 
brain. A given thought, that is to say, is a member of a class, 
and the class is a region in the brain. In this sense, where events 
in brains are concerned, we have no reason to suppose that they 
are not thoughts, but, on the contrary, have strong reason to 
suppose that at least some of them are thoughts. I am using 
“thoughts” as a generic term for mental events. 

When we come to events in parts of physical space-time where 
there are no brains, we have still no positive argument to prove 
that they are not thoughts, except such as may be derived from 
observation of the differences between living and dead matter 
coupled with inferences based on analogy or its absence. We may 
contend, for instance, that habit is in the main confined to living 
matter, and that, since memory is a species of habit, it is unlikely 
that there is memory except where there is living matter. Extend- 
ing this argument, we can observe that the behaviour of living 



matter, especially of its higher forms, is much more dependent 
on its past history than that of dead matter, and that, therefore, 
the whole of that large part of our mental life that depends upon 
habit is presumably only to be found where there is living matter. 
But such arguments are inconclusive and limited in scope. Just 
as we cannot be sure that the sun is not bright, so we cannot be 
sure that it is not intelligent . 1 We may be right in thinking both 
improbable, but we are certainly wrong if we say they are impos- 

I conclude that, while mental events and their qualities can 
be known without inference, physical events are known only 
as regards their space- time structure. The qualities that compose 
such events are unknown — so completely unknown that we 
cannot say either that they are, or that they are not, different 
from the qualities that we know as belonging to mental events. 

1 I do not wish the reader to take this possibility too seriously. It is 
of the order of “pigs might fly”, dealt with by Mr. Crawshay-Williams 
in The Comforts of Unreason , p. 193. 




Chapter I 


I N all that has been said hitherto about the world of science, 
everything has been taken at its face value. I am not saying 
merely that we have taken the attitude of believing what men 
of science tell us, for this attitude, up to a point, is the only 
rational one for any man who is not a specialist on the matter in 
question. In saying that this attitude is rational, I do not mean 
that we should feel sure of the truth of what we are told, for it 
is generally admitted that probably in due course corrections will 
be found necessary. What I do mean is that the best scientific 
opinion of the present time has a better chance of truth, or of 
approximate truth, than any differing hypothesis suggested by 
a layman. The case is analogous to that of firing at a target. If 
you are a bad shot you are not likely to hit the bull's eye, but 
you are nevertheless more likely to hit the bull's eye than to hit 
any other equal area. So the scientist's hypothesis, though 
not likely to be quite right, is more likely to be right than 
any variant suggested by an unscientific person. This, how- 
ever, is not the point with which we are concerned in this 

The matter with which we are now to be concerned is not 
truth, but interpretation. It often happens that we have what seems 
adequate reason to believe in the truth of some formula expressed 
in mathematical symbols, although we are not in a position to 
give a clear definition of the symbols. It happens also, in other 
cases, that we can give a number of different meanings to the 
symbols, all of which will make the formula true. In the former 
case we lack even one definite interpretation of our formula, 
whereas in the latter we have many. This situation, which may 
seem odd, arises in pure mathematics and in mathematical physics; 
it arises even in interpreting common-sense statements such as 
“my room contains three tables and four chairs". It will thus 
appear that there is a large class of statements, concerning each 
of which, in some sense, we are more certain of its truth than of 
its meaning. “Interpretation" is concerned with such statements; 
it consists in finding as precise a meaning as possible for a state- 

25 * 

human knowledge: its scope and limits 

ment of this sort, or, sometimes, in finding a whole system of 
possible meanings. 

Let us take first an illustration from pure mathematics. Mankind 
have long been convinced that 2 + 2 = 4; they have been so 
firmly convinced of this that it has been taken as the stock example 
of something certain. But when people were asked what they 
meant by “2”, “4”, “ + ”, and “ = ”, they gave vague and 
divergent answers, which made it plain that they did not know 
what these symbols meant. Some maintained that we know each 
of the numbers by intuition, and therefore have no need to define 
them. This might seem fairly plausible where small numbers 
were concerned, but who could have an intuition of 3,478,921 ? 
So they said we had an intuition of 1 and +; we could then 
define “2” as “1 + 1”, ”3” as “2 + 1”, “4” as "3 + 1”, and 
so on. But this did not work very well. It enabled us to say that 
2 + 2 = (i + i) + (i + i), and that 4 = {(1 + 1) + 1} + 1 
and we then needed a fresh intuition to tell us that we could 
rearrange the brackets, in fact to assure us that, if /, m y n are 
three numbers, then 

(/ + 77?) + n = l + (m + n) 

Some philosophers were able to produce this intuition on demand, 
but most people remained somewhat sceptical of their claims, and 
felt that some other method was called for. 

A new development, more germane to our problem of inter- 
pretation, was due to Peano. Peano started with three undefined 
terms, “o”, “finite integer (or number)” and “successor of”, and 
concerning these terms he made five assumptions, namely : 

(1) o is a number; 

(2) If a is a number, the successor of a (i.e. a + i) is a number; 

(3) If two numbers have the same successor, the two numbers 

are identical ; 

(4) o is not the successor of any number; 

(5) If s be a class to which belongs o and also the successor 

of every number belonging to s , then every number 
belongs to s. 

The last of these assumptions is the principle of mathematical 



Peano showed that by means of these five assumptions he could 
prove every formula in arithmetic. 

But now a new trouble arose. It was assumed that we need not 
know what we meant by “o”, “number”, and “successor”, so 
long as we meant something satisfying the five assumptions. But 
then it turned out that there were an infinite number of possible 
interpretations. For instance, let “o” mean what we commonly 
call “i”, and let “number” mean what we commonly call “number 
other than o”; then all the five assumptions are still true, and all 
arithmetic can be proved, though every formula will have an 
unexpected meaning. “2” will mean what we usually call “3”, 
but “2 + 2” will not mean “3 + 3”; it will mean “3 + 2”, and 
“2 + 2 = 4” will mean what we usually express by “3 + 2 = 5”. 
In like manner we could interpret arithmetic on the assumption 
that “o” means “100”, and “number” means “number greater 
than 99”. And so on. 

So long as we remain in the region of arithmetical formulae, 
all these different interpretations of “number” are equally good. 
It is only when we come to the empirical uses of numbers in 
enumeration that we find a reason for preferring one interpretation 
to all the others. When we buy something in a shop, and the 
attendant says “three shillings, please”, his “three” is not a mere 
mathematical symbol, meaning “the third term after the beginning 
of some series”; his “three”, in fact, is not capable of being 
defined by its arithmetical properties. It is obvious that his inter- 
pretation of “three” is, outside arithmetic, preferable to all the 
others that Peano’s system leaves possible. Such statements as 
“men have 10 fingers”, “dogs have 4 legs”, “New York has 
10,000,000 inhabitants”, require a definition of numbers which 
cannot be derived from the mere fact that they satisfy the formulae 
of arithmetic. Such a definition is, therefore, the most satisfactory 
“interpretation” of number-symbols. 

The same sort of situation arises whenever mathematics is 
applied to empirical material. Take, for example, geometry, con- 
sidered not as a logical exercise in deducing consequences from 
arbitrarily assumed axioms, but a9 a help in land surveying, map 
making, engineering, or astronomy. Such practical uses of 
geometry involve a difficulty which, though sometimes admitted 
in a perfunctory manner, is never allowed anything like its due 
weight. Geometry, as set forth by the mathematicians, uses points, 


human knowledge: its scope and limits 

lines, planes, and circles, but it is a platitude to say that no such 
objects are to be found in nature. When, in surveying, we use 
the process of triangulation, it is admitted that our triangles do 
not have accurate straight lines for their sides nor exact points 
at their corners, but this is glozed over by saying that the sides 
are approximately straight and the corners approximately points. 
It is not at all clear what this means, so long as it is maintained 
that there are no exact straight lines or points to which our rough' 
and-ready lines and points approximate. We may mean that 
sensible lines and points have approximately the properties set 
forth in Euclid’s definitions and axioms, but unless we can say, 
within limits, how close the approximation is, such a view will 
make calculation vague and unsatisfactory. 

This problem of the exactness of mathematics and the inexact- 
ness of sense is an ancient one, which Plato solved by the fantastic 
hypothesis of reminiscence. In modern times, like some other 
unsolved problems, it has been forgotten through familiarity, like 
a bad smell which you no longer notice because you have lived 
with it so long. It is clear that, if geometry is to be applied to the 
sensible world, we must be able to find definitions of points, lines, 
planes, etc., in terms of sensible data, or else we must be able 
to infer from sensible data the existence of unperceived entities 
having the properties that geometry needs. To find ways, or a 
way, of doing one or other of these things is the problem of the 
empirical interpretation of geometry. 

There is also a non-empirical interpretation, which leaves 
geometry within the sphere of pure mathematics. The assemblage 
of all ordered trios of real numbers forms a three-dimensional 
Euclidean space. With this interpretation, all Euclidean geometry 
is deducible from arithmetic. Every non-Euclidean geometry is 
capable of a similar arithmetical interpretation. It can be proved 
that Euclidean geometry, and every form of non-Euclidean 
geometry, can be applied to every class having the same number 
of terms as the real numbers; the question of the number of 
dimensions, and whether the resulting geometry is Euclidean or 
non-Euclidean, will depend upon the ordering relation that we 
select ; an infinite number of ordering relations exist (in the logical 
sense), and only reasons of empirical convenience can lead us to 
select some one among them for special attention. All this is 
relevant in considering what interpretation of pure geometry had 



better be adopted by the engineer or the physicist. It shows that, 
in an empirical interpretation, the ordering relation, and not only 
the terms ordered, must be defined in empirical terms. 

Very similar considerations apply to time, which, however, so 
far as our present question is concerned, is not so difficult a 
problem as space. In mathematical physics, time is treated as 
consisting of instants, though the perplexed student is assured 
that instants are mathematical fictions. No attempt is made to 
show him why fictions are useful, or how they are related to what 
is not fictitious. He finds that by the use of these fairy tales it is 
possible to calculate what really happens, and after a time he 
probably ceases to trouble himself as to why this is the case. 

Instants were not always regarded as fictions; Newton thought 
them as “real” as the sun and moon. When this view was aban- 
doned, it was easy to swing to the opposite extreme, and to forget 
that a fiction which is useful is not likely to be a mere fiction. 
There are degrees of fictiveness. Let us, for the moment, regard 
an individual person as something in no degree Active ; what, then, 
shall we say of the various aggregates of persons to which he 
belongs? Most people would hesitate to regard a family as a 
fictitious unit, but what about a political party or a cricket club ? 
What about the assemblage of persons called “Smith”, to which 
we will suppose our individual to belong? If you believe in 
astrology, you will attach importance to the assemblage of persons 
born under a certain planet; if you do not, you will regard such 
an assemblage as Active. These distinctions are not logical; from 
the logical point of view, all assemblages of individuals are equally 
real or equally fictive. The importance of the distinctions is 
practical, not logical : there are some assemblages about which there 
are many useful things to be said, and others about which this 
is not the case. 

When we say that instants are useful fictions, we must be 
supposed to mean that there are entities to which, as to individual 
people, we feel inclined to attach a high degree of “reality” 
(whatever that may mean), and that, in comparison with them, 
instants have that lesser degree of “reality” that cricket clubs 
have in relation to their members; but we wish also to say that 
about instants, as about families as opposed to “artificial” aggre- 
gates of people, there are many practically important things to say. 

All this is very vague, and the problem of interpretation is that 



of substituting something precise, remembering always that, 
however we define “instants”, they must have the properties 
required in mathematical physics. Given two interpretations which 
both satisfy this requisite, the choice between them is one of taste 
and convenience; there is not one interpretation which is “right” 
and others that are “wrong”. 

In classical physics, the technical apparatus consists of points, 
instants, and particles. It is assumed that there is a three-term 
relation, that of occupying a point at an instant, and what occupies 
a point at an instant is called a “particle”. It is also assumed 
technically that particles are indestructible, so that whatever 
occupies a point at a given instant occupies some point at every 
other instant. When I say that this is assumed, I do not mean 
that it is asserted to be a fact, but that the technique is based on 
the assumption that no harm will come of treating it as a fact. 
This is still held to be the case in macroscopic physics, but in 
microscopic physics “particles” have been gradually disappearing. 
“Matter” in the old sense is no longer needed; what is needed 
is “energy”, which is not defined except as regards its laws and 
the relation of changes in its distribution to our sensations, more 
especially the relation of frequencies to colour-perceptions. 

Broadly speaking, we may say that the fundamental technical 
apparatus of modern physics is a four-dimensional manifold of 
“events” ordered by space-time relations, which can be analysed 
into a spatial and a temporal component in a number of ways, 
the choice between which is arbitrary. Since the calculus is still 
used, it is still technically assumed that space-time is continuous, 
but it is not clear how far this assumption is more than a mathe- 
matical convenience. Nor is it clear that “events” have that 
precise location in space-time that used to characterize a particle 
at an instant. All this makes the question of the interpretation 
of modern physics very difficult, but in the absence of some 
interpretation we cannot say what is being asserted by the quantum 

“Interpretation”, in its logical aspect, is somewhat different 
from the rather vague and difficult concept which we considered 
at the beginning of this Chapter. We were there concerned with 
symbolic statements which are known to have a connection with 
observable phenomena, and to lead to results which observation 
confirms, but are somewhat indeterminate in meaning except in 


so far as their connection with observation defines them. In this 
case we can say, as we said at the beginning of this Chapter, 
that we are pretty sure our formulae are true, but not at all sure 
what they mean. In logic, however, we proceed differently. Our 
formulae are not regarded as “true” or “false**, but as hypotheses 
containing variables. A set of values of the variables which makes 
the hypotheses true is an “interpretation”. The word “point”, 
in geometry, may be interpreted as meaning “ordered triad of 
real numbers”, or, as we shall see, as meaning what we shall call 
“complete complex of compresence” ; it may also be interpreted 
in an infinite number of other ways. What all the ways have in 
common is that they satisfy the axioms of geometry. 

We often have, both in pure and applied mathematics, collec- 
tions of formulae all logically deducible from a small number of 
initial formulae, which may be called “axioms”. These axioms 
may be regarded as hostages for the whole system, and we may 
concentrate our attention exclusively upon them. The axioms 
consist partly of terms having a known definition, partly of terms 
which, in any interpretation, will remain variables, and partly 
of terms which, though as yet undefined, are intended to acquire 
definitions when the axioms are “interpreted**. The process of 
interpretation consists in finding a constant signification for this 
class of terms. The signification may be given by a verbal defini- 
tion, or may be given ostensively. It must be such that, with this 
interpretation, the axioms become true. (Before interpretation, 
they are neither true nor false.) It thus follows that all their 
consequences are also true. 

Suppose, for example, we wish to interpret the formulae of 
arithmetic. In Peano*s five axioms (given above) there are: first, 
logical terms, such as “is a** and “is identical with**, of which 
the meaning is supposed known ; second, variables, such as a and s, 
which are to remain variables after interpretation; third, the terms 
“o”, “number**, and “successor of**, for which an interpretation 
is to find a constant meaning which makes the five axioms true. 
As we saw, there are an infinite number of interpretations satis- 
fying these conditions, but there is only one among them which 
also satisfies empirical statements of enumeration, such as “I have 
io fingers’*. In this case, therefore, there is one interpretation 
which is very much more convenient than any of the others. 

As we saw in the case of geometry, a given set of axioms may 

257 R 

HUMAN knowledge: its scope and limits 

be capable of two sorts of interpretation, one logical and one 
empirical. All nominal definitions, if pushed back far enough, must 
lead ultimately to terms having only ostensive definitions, and 
in the case of an empirical science the empirical terms must 
depend upon terms of which the ostensive definition is given in 
perception. The astronomer’s sun, for instance, is very different 
from what we see, but it must have a definition derived from the 
ostensive definition of the word “sun” which we learnt in child- 
hood. Thus an empirical interpretation of a set of axioms, when 
complete, must always involve the use of terms which have an 
ostensive definition derived from sensible experience. It will not, 
of course, contain only such terms, for there will always be logical 
terms; but it is the presence of terms derived from experience 
that makes an interpretation empirical. 

The question of interpretation has been unduly neglected. So 
long as we remain in the region of mathematical formulae, every- 
thing appears precise, but when we seek to interpret them it 
turns out that the precision is partly illusory. Until this matter 
has been cleared up, we cannot tell with any exactitude what any 
given science is asserting. 

Chapter II 


I n the present chapter we shall be concerned with a linguistic 
technique which is very useful in the analysis of scientific 
concepts. There are as a rule a number of ways in which the 
words used in a science can be defined in terms of a few among 
them. These few may have ostensive definitions, or may have 
nominal definitions in terms of words not belonging to the science 
in question, or — so long as the science is not “interpreted” in 
the sense considered in the last chapter — they may be left without 
either ostensive or nominal definition, and regarded merely as 
a set of terms having the properties which the science ascribes 
to its fundamental terms. Such a set of initial words I call a 
“minimum vocabulary” for the science in question, provided 
that (a) every other word used in the science has a nominal 
definition in terms of these words, and ( b ) no one of these initial 
words has a nominal definition in terms of the other initial 

Everything said in a science can be said by means of the words 
in a minimum vocabulary. For whenever a word occurs which 
has a nominal definition, we can substitute the defining phrase; 
if this contains words with a nominal definition, we can again 
substitute the defining phrase, and so on, until none of the re- 
maining words have nominal definitions. In fact, definable terms 
are superfluous, and only undefined terms are indispensable. But 
the question which terms are to be undefined is in part arbitrary. 
Take, for example, the calculus of proposition, which is the 
simplest and most completed example of a formal system. We 
can take “or” and “not” as undefined, or “and” and “not”; 
instead of two such undefined terms, we can take one, which 
may be “not this or not that” or “not this and not that”. Thus 
in general we cannot say that such-and-such a word must belong 
to the minimum vocabulary of such-and-such a science, but at 
most that there are one or more minimum vocabularies to which 
it belongs. 

Let us take geography as an example. I shall assume the voca- 
bulary of geometry already established ; then our first distinctively 


HUMAN knowledge: its scope and limits 

geographical need is a method of assigning latitude and longitude. 
For this it will suffice to have as part of our minimum vocabulary 
“Greenwich”, “the North Pole”, and “West of”; but clearly any 
other place would do as well as Greenwich, and the South Pole 
would do as well as the North Pole. The relation “west of” is 
not really necessary, for a parallel of latitude is a circle on the 
earth’s surface in a plane perpendicular to the diameter passing 
through the North Pole. The remainder of the words used in 
physical geography, such as “land” and “water”, “mountain” 
and “plain”, can now be defined in terms of chemistry, physics, 
or geometry. Thus it would seem that it is the two words “Green- 
wich” and “North Pole” that are needed in order to make 
geography a science concerning the surface of the earth, and not 
some other spheroid. It is owing to the presence of these two 
words (or two others serving the same purpose) that geography 
is able to relate the discoveries of travellers. It is to be observed 
that these two words are involved wherever latitude and longitude 
are mentioned. 

As this example illustrates, a science is apt to acquire a smaller 
minimum vocabulary as it becomes more systematic. The ancients 
knew many geographical facts before they knew how to assign 
latitudes and longitudes, but to express these facts they needed 
a larger number of undefined words than we need. Since the 
earth is a spheroid, not a sphere, “North-Pole” need not be 
undefined : we can define the two Poles as the extremities of the 
earth’s shortest diameter, and the North Pole as the Pole nearer 
to Greenwich. In this way we can manage with “Greenwich” as 
the only undefined term peculiar to geography. The earth itself 
is defined as “that spheroid whose surface is formed of land and 
water bounded by air, and on whose surface Greenwich is 
situated”. But here we seem to reach a dead end in the way of 
diminishing our minimum vocabulary: if we are to be sure that 
we are talking about the earth, we must mention some place on 
its surface or having a given geometrical relation to it, and the 
place must be one which we can recognize. Therefore although 
“New York” or “Moscow” or “Timbuctoo” would do just as 
well as “Greenwich”, some place must be included in any minimum 
vocabulary for geography. 

One further point is illustrated by our discussion of Greenwich, 
and that is, that the terms which are officially undefined in a 



science may not be identical with those that are undefined for 
a given person. If you have never seen Greenwich, the word 
“Greenwich” cannot, for you, have an ostensive definition; there- 
fore you cannot understand the word unless it has a nominal 
definition. In fact, if you live in a place called “P”, then for you 
“P” takes the place of Greenwich, and your official longitude, 
for you, defines the meridian of Greenwich, not the longitude 
of P. Such considerations, however, are pre-scientific, and are 
usually ignored in the analysis of scientific concepts. For certain 
purposes, they cannot be ignored, particularly when we are con- 
sidering the relation of science to sensible experience; but as a 
rule there is little danger in ignoring them. 

Let us consider next the question of minimum vocabularies 
for astronomy. Astronomy consists of two parts, one a kind of 
cosmic geography, the other an application of physics. Statements 
as to the size and orbits of the planets belong to cosmic geography, 
whereas Newton’s and Einstein’s theories of gravitation belong 
to physics. The difference is that, in the geographical part, we 
are concerned with statements of fact as to what is where, while 
in the part which is physics we are concerned with laws. As I 
shall presently be considering physics on its own account, let us 
consider first the geographical part of astronomy. In this part, 
so long as it is in an elementary stage, we need proper names 
for the sun, the moon, the planets, and all the stars and 
nebulae. The number of proper names required can, however, 
be steadily reduced as the science of astronomy advances. 
“Mercury” can be defined as meaning “the planet nearest the 
sun”, “Venus” as “the second planet”, “the earth” as “the 
third planet”, and so on. Constellations are defined by their 
co-ordinates, and the several stars in a constellation by their 
order of brightness. 

On this system, “the sun” will remain part of our minimum 
vocabulary, and we shall need what is necessary for defining 
celestial co-ordinates. “The Pole Star” will not be necessary, 
since it may be defined as “the star without diurnal revolution”, 
but we shall need some other heavenly body to fulfil the function 
which Greenwich fulfils in terrestrial geography. In this way 
official astronomy could get on with (it would seem) only two 
proper names, “the sun” and (say) “Sirius”. “The moon”, for 
instance, can be defined as “the body whose co-ordinates on such- 



and-such a date are so-and-so.” With this vocabulary we can, 
in a sense, state everything that the astronomer wishes to say, 
just as, with Peano’s three undefined terms, we can state all 

But just as Peano’s system proves inadequate when we come 
to counting, so our official astronomy proves inadequate when 
we attempt to link it to observation. There are two essential 
propositions which it fails to include, namely “that is the sun” 
and “that is Sirius”. We have, it appears, formed a vocabulary 
for astronomy in the abstract, but not for astronomy as a record 
of observations. 

Plato, who was interested in astronomy solely as a body of 
laws, wished it to be wholly divorced from sense; those who were 
interested in the actual heavenly bodies that happen to exist 
would, he said, be punished in the next incarnation by being 
birds. This point of view is not nowadays adopted by men of 
science, but it, or something very like it, is to be found in the 
works of Carnap and some other logical positivists. They are 
not, I think, conscious of holding any such opinion, and 
would vehemently repudiate it; but absorption in words, 
as opposed to what they mean, has exposed them to Platonic 
temptation, and led them down strange paths towards perdition, 
or what an empiricist must consider such. Astronomy is not 
merely a collection of words and sentences; it is a collection of 
words and sentences chosen, from others that were linguistically 
just as good, because they described a world connected with 
sensible experience. So long as sensible experience is ignored, 
no reason appears for concerning ourselves with a large body 
having just so many planets at just such distances from it. And 
the sentences in which sensible experience breaks in are such 
as “that is the sun”. 

Every advanced empirical science has two aspects: on the one 
hand, it consists of a body of propositions interconnected in 
various ways, and often containing a small selection from which 
all the others can be deduced ; on the other hand, it is an attempt 
to describe some part or aspect of the universe. In the former 
aspect, the truth or falsehood of the several propositions is not 
in question, but only their mutual connections. For example, if 
gravitation varied directly as the distance, planets (if any) would 
revolve round the sun (if it existed) in ellipses of which the sun 



would occupy the centre, not a focus. This proposition is not 
part of descriptive astronomy. There is a similar statement, also 
not part of descriptive astronomy, saying that if gravitation varies 
inversely as the square of the distance, planets (if any) will go 
round the sun (if any) in ellipses of which the sun will occupy 
a focus. This is different from the two statements: gravitation 
varies inversely as the square of the distance, and planets revolve 
in ellipses round the sun in a focus. The former statement is a 
hypothetical; the two latter assert both the antecedent and the 
consequent of the previous hypothetical. What enables them to 
do this is the appeal to observation. 

The appeal to observation is made in statements such as “that 
is the sun”; such statements, therefore, are essential to the truth 
of astronomy. Such statements never appear in any finished 
exposition of an astronomical theory, but they do appear while 
a theory is being established. For instance, after the eclipse 
observations in 1919, we were told that the photographs of certain 
stars appeared with such-and-such a displacement towards the 
sun. This was a statement as to the positions of certain dots on 
a photographic plate, as observed by certain astronomers at a 
certain date; it was a statement not primarily belonging to 
astronomy, but to biography, and yet it constituted the evidence 
for an important astronomical theory. 

The vocabulary of astronomy, it thus appears, is wider if we 
consider it as a body of propositions deriving truth, or at least 
probability, from observation, than it is if we treat it as a purely 
hypothetical system whose truth or falsehood does not concern 
us. In the former case we must be able to say “that is the sun”, 
or something of the sort; in the latter case, no such necessity 

Physics, which we must next consider, is in a different position 
from geography and astronomy, since it is not concerned to say 
what exists where, but only to establish general laws. “Copper 
conducts electricity” is a law of physics, but “there is copper 
in Cornwall” is a fact of geography. The physicist as such does 
not care where there is copper, so long as there is enough in his 

In the earlier stages of physics the word “copper” was neces- 
sary, but now it has become definable. “Copper” is “the element 
whose atomic number is 29”, and this definition enables us to 



deduce many things about the copper atom. All the elements can 
be defined in terms of electrons and protons, or at any rate of 
electrons, positrons, neutrons, and protons. (Perhaps a proton 
consists of a neutron and a positron.) These units themselves 
can be defined by their mass and electric charge. In the last 
analysis, since mass is a form of energy, it would seem that energy, 
electric charge, and space-time co-ordinates are all that physics 
needs ; and owing to the absence of the geographical element the 
co-ordinates can remain purely hypothetical, i.e. there need be 
no analogue of Greenwich. Physics as a “pure” science — i.e. apart 
from methods of verification — would seem, therefore, to require 
only a four-dimensional continuum containing distributions of 
varying amounts of energy and electricity. Any four-dimensional 
continuum will do, and “energy” and “electricity” need only be 
quantities whose mode of change of distribution is subject to 
certain assigned laws. 

When physics is brought to this degree of abstraction it becomes 
a branch of pure mathematics, which can be pursued without 
reference to the actual world, and which requires no vocabulary 
beyond that of pure mathematics. The mathematics, however, 
are such as no pure mathematician would have thought of for 
himself. The equations, for instance, contain Planck’s constant A, 
of which the magnitude is about 6*55 X io“ 27 erg secs. No one 
would have thought of introducing just this quantity if there had 
not been experimental reasons for doing so, and as soon as we 
introduce experimental reasons the whole picture is changed. The 
four-dimensional continuum is no longer a mere mathematical 
hypothesis, but the space-time continuum to which we have been 
led by successive refinements of the space and time with which 
we are familiar in experience. Electricity is no longer just any 
quantity, but the thing measured by the observable behaviour 
of our electrical instruments. Energy, though highly abstract, is 
a generalization arrived at by means of completely concrete 
experiments such as those of Joule. Physics as verifiable, therefore, 
uses various empirical concepts in addition to those purely abstract 
concepts that are needed in “pure” physics. 

Let us consider in more detail the definition of such a term 
as “energy”. The important point about energy is its constancy, 
and the chief step in establishing its constancy was the deter- 
mination of the mechanical equivalent of heat. This was effected 



by observation, for example of thermometers. If, then, we mean 
by “physics'* not merely the body of physical laws, but these 
together with the evidence for their truth, then we must include 
in “physics” Joule's perceptions when he looked at thermometers. 
And what do we mean by “heat”? The plain man means a certain 
kind of sensation, or its (to him) unknown cause; the physicist 
means a rapid agitation of the minute parts of bodies. But what 
has led the physicist to this definition? Only the fact that, when 
we feel heat, there is reason to think that such agitation is occur- 
ring. Or take the fact that friction causes heat: our primary 
evidence for this fact is that when we have seen friction we can 
feel heat. All the non-mathematical terms used in physics con- 
sidered as an experimental science have their origin in our sensible 
experience, and it is only on this account that sensible experience 
can confirm or confute physical laws. 

It thus appears that, if physics is regarded as a science based 
on observation, not as a branch of pure mathematics, and if the 
evidence for physical laws is held to be part of physics, then any 
minimum vocabulary for physics must be such as to enable us 
to mention the experiences upon which our physical beliefs are 
based. We shall need such words as “hot”, “red”, “hard”, not 
only to describe what physics asserts to be the condition of bodies 
that give us these sensations, but also to describe the sensations 
themselves. Suppose I say, for instance: By “red” light I mean 
light of such-and-such a range of wave-lengths. In that case the 
statement that light of such wave-lengths makes me see red is 
a tautology, and until the nineteenth century people were uttering 
meaningless noises when they said that blood is red, because 
nothing was known of the correlation of wave-lengths with sen- 
sations of colour. This is absurd. It is obvious that “red” has a 
meaning independent of physics, and that this meaning is relevant 
in collecting data for the physical theory of colours, just as the 
pre-scientific meaning of “hot” is relevant in establishing the 
physical theory of heat. 

The main conclusion of the above discussion of minimum 
vocabularies is that every empirical science, however abstract, 
must contain in any minimum vocabulary words descriptive of 
our experiences. Even the most mathematical terms, such as 
“energy”, must, when the chain of definitions is completed until 
we reach terms of which there is only an ostensive definition, 


HUMAN knowledge: its scope and limits 

be found to depend for their meaning upon terms directly 
descriptive of experiences, or even, in what may be called the 
“g e °g ra phi c al” sciences, giving names to particular experiences. 
This conclusion, if valid, is important, and affords great assistance 
in the work of interpreting scientific theories. 


Chapter III 

I n the present chapter we shall be concerned with a purely 
logical discussion which is essential as a preliminary to any 
further steps in the interpretation of science. The logical 
concept which I shall endeavour to explain is that of “structure”. 

To exhibit the structure of an object is to mention its parts 
and the ways in which they are interrelated. If you were learning 
anatomy, you might first learn the names and shapes of the various 
bones, and then be taught where each bone belongs in the skeleton. 
You would then know the structure of the skeleton in so far as 
anatomy has anything to say about it. But you would not have 
come to an end of what can be said about structure in relation 
to the skeleton. Bones are composed of cells, and cells of mole- 
cules, and each molecule has an atomic structure which it is the 
business of chemistiy to study. Atoms, in turn, have a structure 
which is studied in physics. At this point orthodox science ceases 
its analysis, but there is no reason to suppose that further analysis 
is impossible. We shall have occasion to suggest the analysis of 
physical entities into structures of events, and even events, as 
I shall try to show, may be regarded with advantage as having 
a structure. 

Let us consider next a somewhat different example of structure, 
namely sentences. A sentence is a series of words, arranged in 
order by the relation of earlier and later if the sentence is spoken, 
and of left to right if it is written. But these relations are not 
really between words; they are between instances of words. A 
word is a class of similar noises, all having the same meaning 
or nearly the same meaning. (For simplicity I shall confine myself 
to speech as opposed to writing.) A sentence also is a class of 
noises, since many people can utter the same sentence. We must 
say, then, not that a sentence is a temporal series of words, but 
that a sentence is a class of noises, each consisting of a series of 
noises in quick temporal succession, each of these latter noises 
being an instance of a word. (This is a necessary but not a sufficient 
characteristic of a sentence ; it is not sufficient because some series 
of words are not significant.) I will not linger on the distinction 



between different parts of speech, but will go on to the next stage 
in analysis, which belongs no longer to syntax, but to phonetics. 
Each instance of a word is a complex sound, the parts being the 
separate letters (assuming a phonetic alphabet). Behind the 
phonetic analysis there is a further stage: the analysis of the 
complex physiological process of uttering or hearing a single letter. 
Behind the physiological analysis is the analysis of physics, and 
from this point onward analysis proceeds as in the case of the 

In the above account I passed hastily over two points that need 
elucidation, namely that words have meaning and sentences have 
significance. “Rain” is a word, but “raim” is not, though both are 
classes of similar noises. “Rain is falling” is a sentence, but “rain 
snow elephant” is not, though both are series of words. To define 
“meaning” and “significance” is not easy, as we saw in discussing 
the theory of language. The attempt is not necessary so long as 
we confine ourselves strictly to questions of structure. A word 
acquires meaning by an external relation, just as a man acquires 
the property of being an uncle. No post mortem, however 
thorough, will reveal whether the man was an uncle or not, and 
no analysis of a set of noises (so long as everything external to 
them is excluded) will show whether this set of noises has 
meaning, or significance if the set is a series of what seem to 
be words. 

, The above example illustrates that an analysis of structure, 
however complete, does not tell you all that you may wish to 
know about an object. It tells you only what are the parts of the 
object and how they are related to each other ; it tells you nothing 
about the relations of the object to objects that are not parts or 
components of it. 

The analysis of structure usually proceeds by successive stages, 
as in both the above examples. What are taken as unanalysed 
units in one stage are themselves exhibited as complex structures 
in the next stage. The skeleton is composed of bones, the bones 
of cells, the cells of molecules, the molecules of atoms, the atoms 
of electrons, positrons, and neutrons; further analysis is as yet 
conjectural. Bones, molecules, atoms, and electrons may each be 
treated, for certain purposes, as if they were unanalysable units 
devoid of structure, but at no stage is there any positive reason 
to suppose that this is in fact the case. The ultimate units 



so far reached may at any moment turn out to be capable of 
analysis. Whether there must be units incapable of analysis 
because they are destitute of parts, is a question which there 
seems no way of deciding. Nor is it important, since there is 
nothing erroneous in an account of structure which starts from 
units that are afterwards found to be themselves complex. For 
example, points may be defined as classes of events, but that does 
not falsify anything in traditional geometry, which treated points 
as simples. Every account of structure is relative to certain units 
which are, for the time being, treated as if they were devoid of 
structure, but it must never be assumed that these units will not, 
in another context, have a structure which it is important to 

There is a concept of “identity of structure” which is of great 
importance in relation to a large number of questions. Before 
giving a precise definition of this concept I will give some pre- 
liminary illustrations of it. 

Let us begin with linguistic illustrations. Suppose that, in any 
given sentence, you substitute other words, but in a way which 
still leaves the sentence significant; then the new sentence has 
the same structure as the original one. Suppose, e.g., you start 
with “Plato loved Socrates”; for “Plato” substitute “Brutus”, for 
“loved” substitute “killed”, and for “ Socrates” substitute “Caesar”. 
You thus arrive at the sentence “Brutus killed Caesar”, which has 
the same structure as “Plato loved Socrates”. All sentences having 
this structure are called “dyadic-relation sentences”. Similarly 
from “Socrates is Greek” you could have passed to “Brutus is 
Roman” without change of structure; sentences having this 
structure are called “subject-predicate sentences”. In this way 
sentences can be classified by their structure ; there are in theory 
an infinite number of structures that sentences may have. 

Logic is concerned with sentences that are true in virtue of 
their structure, and that always remain true when other words are 
substituted, so long as the substitution does not destroy signifi- 
cance. Take, for example, the sentence: “If all men are mortal 
and Socrates is a man, then Socrates is mortal.” Here we may 
substitute other words for “Socrates”, “man”, and “mortal”, 
without destroying the truth of the sentence. It is true that there 
are other words in the sentence, namely “if-then” (which must 
count as one word), “all”, “are”, “and’*, “is”, “a”. These words 


HUMAN knowledge: its scope and limits 

must not be changed. But these are “logical” words, and their 
purpose is to indicate structure; when they are changed, the 
structure is changed. (All this raises problems, but it is not 
necessary for our present purpose to go into them.) A sentence 
belongs to logic if we can be sure that it is true (or that it is false) 
without having to know the meanings of any of the words except 
those that indicate structure. That is the reason for the use of 
variables. Instead of the above sentence about Socrates and man 
and mortal, we say: “If all cl's are /?’s and x is an a, then x is a j8.” 
Whatever x and a and ft may be, this sentence is true; it is true 
in virtue of its structure. It is in order to make this clear that we 
use and “a” and “0” instead of ordinary words. 

Let us take next the relation of a district to a map of it. If the 
district is small, so that the curvature of the earth can be neglected, 
the principle is simple : east and west are represented by right and 
left, north and south by up and down, and all distances are reduced 
in the same proportion. It follows that from every statement about 
the map you can infer one about the district, and vice versa. If 
there are two towns, A and B, and the map is on the scale of an 
inch to the mile, then from the fact that the mark “A” is ten 
inches from the mark “B” you can infer that A is ten miles from 
B, and conversely; and from the direction of the line from the 
mark “A” to the mark “B” you can infer the direction of the line 
from A to B. These inferences are possible owing to identity of 
structure between the map and the district. 

Now take a somewhat more complicated illustration: the 
relation of a gramophone record to the music that it plays. It is 
obvious that it could not produce this music unless there were a 
certain identity of structure between it and the music, which can 
be exhibited by translating sound-relations into space-relations, 
or vice versa — e.g. what is nearer to the centre on the record 
corresponds to what is later in time in the music. It is only because 
of the identity of structure that the record is able to cause the 
music. Very similar considerations apply to telephones, broad- 
casting, etc. 

We can generalize such examples so as to deal with the rela- 
tions of our perceptual experiences to the external world. A 
wireless set transforms electromagnetic waves into sound waves; 
a human organism transforms sound waves into auditory sensa- 
tions. The electromagnetic waves and the sound waves have a 



certain similarity of structure, and so (we may assume) have the 
sound waves and the auditory sensations. Whenever one complex 
structure causes another, there must be much the same structure 
in the cause and in the effect, as in the case of the gramophone 
record and the music. This is plausible if we accept the maxim 
“same cause, same effect” and its consequence, “different effects, 
different causes”. If this principle is regarded as valid, we can 
infer from a complex sensation or series of sensations the structure 
of its physical cause, but nothing more, except that relations of 
neighbourhood must be preserved, i.e. neighbouring causes have 
neighbouring effects. This argument is one which needs much 
amplification ; for the moment I am merely mentioning it by way 
of anticipation, in order to show one of the important applications 
of the concept of structure. 

We can now proceed to the formal definition of “structure”. 
It is to be observed that structure always involves relations: a 
mere class, as such, has no structure. Out of the terms of a given 
class many structures can be made, just as many different sorts of 
houses can be made out of a given heap of bricks. Every relation 
has what is called a “field”, which consists of all the terms that 
halve the relation to something or to which something has the 
re ation. Thus the field of “parent” is the class of parents and 
children, and the field of “husband” is the class of husbands and 
wives. Such relations have two terms, and are called “dyadic”. 
There are also relations of three terms, such as jealousy and 
“between”; these are called “triadic”. If I say “A bought B from 
C for D pounds”, I am using a “tetradic” relation. If I say “A 
minds B’s love for C more than D’s hatred of E”, I am using a 
“pentadic” relation. To this series of kinds of relation there is no 
theoretical limit. 

Let us in the first instance confine ourselves to dyadic relations. 
We shall say that a class a ordered by the relation R has the same 
structure as a class £ ordered by the relation S, if to every term 
in a some one term in fi corresponds, and vice versa, and if when 
two terms in a have the relation R, then the corresponding terms 
in /? have the relation S, and vice versa. We may illustrate by the 
similarity between a spoken and a written sentence. Here the class 
of spoken words in the sentence is a, the class of written words in 
the sentence is j8, and if one spoken word is earlier than another, 
then the written word corresponding to the one is to the left of 


HUMAN knowledge: its scope and limits 

the written word corresponding to the other (or to the right if the 
language is Hebrew). It is in consequence of this identity of 
structure that spoken and written sentences can be translated into 
each other. The process of learning to read and write is the process 
of learning which spoken word corresponds to a given written 
word and vice versa. 

A structure may be defined by several relations. Take, for 
instance, a piece of music. One note may be earlier or later than 
another, or simultaneous with it. One note may be louder than 
another, or higher in pitch, or differing through a wealth or 
poverty of harmonics. All the relations of this kind that are 
musically relevant must have analogues in a gramophone record 
if it is to give a good reproduction. In saying that the record must 
have the same structure as the music, we are not concerned with 
only one relation R between the notes of the music and one 
corresponding relation S between the corresponding marks on 
the record, but with a number of relations such as R and a number 
of corresponding relations such as S. Some maps use different 
colours for different altitudes; in that case, different positions on 
the map correspond to different latitudes and longitudes, while 
different colours correspond to different elevations. The identity 
of structure in such maps is greater than in others; that is why 
they are able to give more information. 

The definition of identity of structure is exactly the same for 
relations of higher orders as it is for dyadic relations. Given, for 
example, two triadic relations R and S, and given two classes 
a and /? of which a is contained in the field of R while j8 is con- 
tained in the field of S, we shall say that a ordered by R has the 
same structure as 0 ordered by S if there is a way of correlating 
one member of a to one of /J, and vice versa, so that, if a v a 2 , a 3 
are correlated respectively with b v b 2> fi 3 , if R relates a v a 2f a 3 
(in that order), then S relates b v b 2 , b 3 (in that order), and vice 
versa. Here, again, there may be several relations such as R, and 
several such as S; in that case, there is identity of structure in 
various respects. 

When two complexes have the same structure,* every statement 
about the one, in so far as it depends only on structure, has a 
corresponding statement about the other, true if the first was true, 
and false if the first was false. Hence arises the possibility of a 
dictionary, by means of which statements about the one complex 



can be translated into statements about the other. Or, instead of a 
dictionary, we may continue to use the same words, but attach 
different meanings to them according to the complex to which 
they are referred. This sort of thing happens in interpreting a 
sacred text or the laws of physics. The “days” in the Biblical 
account of the Creation are taken to mean “ages”, and in this way 
Genesis is reconciled with geology. In physics, assuming that our 
knowledge of the physical world is only as to the structure resulting 
from the empirically known relation of “neighbourhood” in the 
topological sense, we have immense latitude in the interpretation 
of our symbols. Every interpretation that preserves the equations 
and the connection with our perceptive experiences has an equal 
claim to be regarded as possibly the true one, and may be used with 
equal right by the physicist to clothe the bare bones of his mathe- 

Take, for example, the question of waves versus particles. Until 
recently it was thought that this was a substantial question : light 
must consist either of waves or of little packets called photons. 
It was regarded as unquestionable that matter consisted of 
particles. But at last it was found that the equations were the same 
if both matter and light consisted of particles, or if both consisted 
of waves. Not only were the equations the same, but all the 
verifiable consequences were the same. Either hypothesis, there- 
fore, is equally legitimate, and neither can be regarded as having 
a superior claim to truth. The reason is that the physical world 
can have the same structure, and the same relation to experience, 
on the one hypothesis as on the other. 

Considerations derived from the importance of structure show 
that our knowledge, especially in physics, is much more abstract 
and much more infected with logic than it used to seem. There is 
however a very definite limit to the process of turning physics into 
logic and mathematics; it is set by the fact that physics is an 
empirical science, depending for its credibility upon relations to 
our perceptive experiences. The further development of this 
theme must be postponed until we come to the theory of scientific 

2 73 

Chapter IV 


T he reader will remember that, in relation to a given body 
of knowledge, a minimum vocabulary is defined as one 
having the two properties (i) that every proposition in the 
given body of knowledge can be expressed by means of words 
belonging to the minimum vocabulary, (2) that no word in this 
vocabulary can be defined in terms of other words in it. In the 
present chapter I wish to show the connection of this definition 
with structure. 

The first thing to notice is that a minimum vocabulary cannot 
contain names for complexes of which the structure is known. 
Take (say) the name “France”. This denotes a certain geographical 
region, and can be defined as “all places within such-and-such 
boundaries”. But we cannot conversely define the boundaries in 
terms of “France”. We want to be able to say “this place is on 
the boundaries of France”, which requires a name for this place, 
or for constituents which compose it. “This place” enters into 
the definition of “France”, but “France” does not enter into the 
definition of “this place”. 

It follows that every discovery of structure enables us to 
diminish the minimum vocabulary required for a given subject- 
matter. Chemistry used to need names for all the elements, but 
now the various elements can be defined in terms of atomic 
structure, by the use of two words, “electron” and “proton” 
(or perhaps three words, “electron”, “positron”, and “neutron”). 
Any region in space-time can be defined in terms of its parts, but 
its parts cannot be defined in terms of it. A man can be defined 
by enumerating, in the right temporal order, all the events that 
happen to him, but the events cannot be defined in terms of him. 
If you wish to speak both about complexes and about the things 
that are in fact their constituents, you can always achieve it with- 
out names for the complexes, if you know their structure. In this 
way analysis simplifies, systematizes, and diminishes your initial 

The words required in an empirical science are of three sorts. 
There are, first, proper names, which usually denote some con- 



tinuous portion of space-time; such are “Socrates”, “Wales”, 
“the sun”. Then there are words denoting qualities or relations; 
instances of qualities are “red”, “hot”, “loud”, and instances of 
relations are “above”, “before”, “between”. Then there are 
logical words, such as “or”, “not”, “some”, “all”. For our present 
purposes we may ignore logical words, and concentrate upon the 
other two kinds. 

It is usually taken for granted that the analysis of something 
that has a proper name consists in dividing it into spatio-temporal 
parts. Wales consists of counties, the counties consist of parishes, 
each parish consists of the church, the school, etc. The church 
in turn has parts, and so we can continue (it is thought) until we 
reach points. The odd thing is that we never do reach points, and 
that the familiar building thus seems to be composed of an 
infinity of unattainable and purely conceptual constituents. I 
believe this view of spatio-temporal analysis to be mistaken. 

Qualities and relations are sometimes analysable, sometimes 
not. I do not believe that “before”, as we know it in experience, 
can be analysed ; at any rate I do not know any analysis of it that 
I am willing to accept. But in some cases the analysis of a relation 
is obvious. “Grandparent” means “parent of parent”, “brother” 
means “son of parent”, and so on. All family relationships can be 
expressed by means of the three words “spouse”, “male”, and 
“parent”; this is a minimum vocabulary in this subject-matter. 
Adjectives (i.e. words denoting qualities) are often complex in 
their meaning. Milton calls the woodbine “well-attired”, which 
is a word of which the meaning is very complex. So is such a word 
as “famous”. Words such as “red”, which come nearer to sim- 
plicity, do not achieve it; there are many shades of red. 

Whenever the analysis of a quality or relation is known, the 
word for that quality or relation is unnecessary in our “basic 

When we have words for every thing, quality, and relation that 
we cannot analyse, we can express all our knowledge without 
the need of any other words. In practice this would be too lengthy, 
but in theory nominal definitions are unnecessary. 

If the world is composed of simples, i.e. of things, qualities, 
and relations that are devoid of structure, then not only all our 
knowledge, but all that of Omniscience, could be expressed by 
means of words denoting these simples. We could distinguish in 


HUMAN knowledge: its scope and limits 

the world a stuff (to use William James’s word) and a structure. 
The stuff would consist of all the simples denoted by names, 
while the structure would depend on relations and qualities for 
which our minimum vocabulary would have words. 

This conception can be applied without assuming that there is 
anything absolutely simple. We can define as “relatively simple” 
whatever we do not know to be complex. Results obtained by 
using the concept of “relative simplicity” will still be true if 
complexity is afterwards found, provided we have abstained from 
asserting absolute simplicity. 

If we allow denotative as opposed to structural definitions, we 
can, at least apparently, content ourselves with a much smaller 
apparatus of names. All places in space-time can be indicated by 
their co-ordinates, all colours by their wave-lengths, and so on. 
We have already seen that the assignment of space- time co-ordi- 
nates requires a few proper names, say “Greenwich”, “the Pole 
Star”, and “Big Ben”. But this is a very small apparatus compared 
to names for all the different places in the universe. Whether this 
way of defining spatio-temporal places enables us to say all that 
we know about them, is a difficult question, to which I shall return 
shortly. Before discussing it, it will be well to examine more 
closely the questions that arise concerning qualities. 

Consider the definition of the word “red”. We may define it 
(i) as any shade of colour between two specified extremes in the 
spectrum, or (2) as any shade of colour caused by wave-lengths 
lying between specified extremes, or (3) (in physics) as waves 
having wave-lengths between these extremes. There are different 
things to be said about these three definitions, but there is one 
thing to be said about all of them. 

What is to be said about all of them is that they have an artificial, 
unreal, and partly illusory precision. The word “red”, like the 
word “bald”, is one which has a meaning that is vague at the edges. 
Most people would admit that, if a man is not bald, the loss of 
one hair will not make him so ; it follows by mathematical induc- 
tion that the loss of all his hairs will not make him so, which is 
absurd. Similarly, if a shade of colour is red, a very tiny change 
will not make it cease to be red, from which it follows that all 
shades of colour are red. The same sort of thing happens when 
we use wave-lengths in our definition, since lengths cannot 
be accurately measured. Given a length which, by the most 



careful measurements, appears to be a metre, it will still appear 
to be a metre if it is very slightly increased or diminished; there- 
fore every length appears to be a metre, which again is absurd. 

It follows from these considerations that any definition of “red” 
which professes to be precise is pretentious and fraudulent. 

We shall have to define “red”, or any other vague quality, by 
some such method as the following. When the colours of the 
spectrum are spread out before us, there are some that everybody 
would agree to be red, and others that everybody would agree to 
be not red, but between these two regions of the spectrum there 
is a doubtful region. As we travel along this region, we shall begin 
by saying “I am nearly certain that that is red”, and end by saying 
“I am nearly certain that that is not red”, while in the middle 
there will be a region where we have no preponderant inclination 
either towards yes or towards no . All empirical concepts have this 
character — not only obviously vague concepts such as “loud” or 
“hot”, but also those that we are most anxious to make precise, 
such as “centimetre” and “second”. 

It might be thought that we could make “red” precise by 
confining the term to those shades that we are certain are red. 
This, however, though it diminishes the area of uncertainty, does 
not abolish it. There is no precise point in the spectrum where 
you are sure that you become uncertain. There will still be three 
regions, one where you are certain that you are certain the shade 
is red, one where you are certain that you are uncertain, and an 
intermediate region where you are uncertain as to whether you 
are certain or uncertain. And these three regions, like the previous 
ones, will have no sharp boundaries. You have merely adopted 
one of the innumerable techniques which diminish the area of 
vagueness without ever wholly abolishing it. 

The above discussion has proceeded on the assumption of 
continuity. If all change is discrete — and we do not know that it 
is not — then complete accuracy is theoretically possible. But if 
there is discontinuity it lies, for the present, far below the level 
of sensible discrimination, so that discreteness, even if it should 
exist, would be useless as a help in defining empirically given 

Let us now ignore the problem of vagueness, and revert to our 
three definitions. But we will now adapt them so as to be defini- 
tions of a given shade of colour. This introduces no new diffi- 



culties, since, as we have seen, the definition of “red” as a band 
of colours requires a definition of the precise shades that form 
its boundaries. 

Let us suppose that I am seeing a certain coloured patch, and 
that I call the shade of the patch “C”. Physics tells me that this 
shade of colour is caused by light of wave-length A. I may then 
define “C” as: (i) the shade of any patch that is indistinguishable 
in colour from the patch I am seeing now ; or (2) as the shade of 
any visual sensation caused by electromagnetic waves of wave- 
length A; or (3) as electromagnetic waves of wave-length A. When 
we are concerned only with physics, without regard to the methods 
by which its laws are verified, (3) is the most convenient definition. 
We use it when we speak of ultra-violet light, and when we say 
that the light from Mars is red, and when, during a sunset, we 
say that the sun’s light is not really red, but only looks red because 
of intervening mist. Physics, per se y has nothing to say about 
sensations, and if it uses the word “colour” (which it need not do), 
it will wish to define it in a way that is logically independent of 

But although physics as a self-contained logical system does 
not need to mention sensations, it is only through sensations that 
physics can be verified. It is an empirical law that light of a certain 
wave-length causes a visual sensation of a certain kind, and it is 
only when such laws are added to those of physics that the total 
becomes a verifiable system. The definition (2) has the defect of 
concealing the force of the empirical law which connects wave- 
length with sensation. Names for colours were used for thousands 
of years before the undulatory theory of light was invented, and it 
was a genuine discovery that wave-lengths grow shorter as we 
travel along the spectrum from red to violet. If we define a shade 
of colour by its wave-length, we shall have to add that sensations 
caused by light of the same wave-length all have a recognizable 
similarity, and that there is a lesser degree of similarity when the 
wave-lengths differ, but only by a little. Thus we cannot express 
all that we know on the subject without speaking about shades of 
colour as known directly in visual sensation, independently of any 
physical theory as to light-waves. 

It would seem, therefore, that if we wish for clarity in exhibiting 
the empirical data which lead us to accept physics, we shall do well 
to adopt our first definition of a shade of colour, since we shall 



certainly need some way of speaking about what this definition 
defines, without having to make the detour through physics that 
is involved in mentioning wave-lengths. 

It remains, however, an open question whether the raw material 
in our definitions of colours should be a given shade of colour 
(wherever it may occur), or a given patch of colour, which can 
only occur once. Let us develop both hypotheses. 

Suppose I wish to give an account of my own visual field 
throughout a certain day. Since we are concerned only with 
colour, depth may be ignored. I have therefore at each moment 
a two-dimensional manifold of colours. I shall assume that my 
visual field can be divided into areas of finite size, within each 
of which the colour is sensibly uniform. (This assumption is not 
essential, but saves verbiage.) My visual field, on this assumption, 
will consist of a finite number of coloured patches of varying 
shape. I may start by giving a name to each patch, or by giving 
a name to each shade of colour. We have to consider whether 
there are any reasons for preferring one of these courses to the 

If I start by giving a name to each patch, I proceed to the 
definition of a shade of colour by means of a relation of colour- 
similarity between patches. This similarity may be greater or less; 
we suppose that there is an extreme degree of it which may be 
called “exact likeness”. This relation is distinguished by being 
transitive, which is not the case with minor degrees of resem- 
blance. For the reasons already given, we can never be sure that, 
in any given case, there is exact colour-likeness between two 
patches, any more than we can be sure that a given length is 
exactly a metre. However, we can invent techniques which 
approximate more and more closely to what would be needed for 
establishing exact likeness. 

We define the shade of colour of a given patch as the class of 
patches having exact colour-likeness to it. Every shade of colour 
is defined in relation to a “this”; it is “the shade of colour of this 
patch”. To each “this”, as we become aware of it, we give a name, 
say “P”; then “the shade of P” is defined as “all patches having 
exact colour-likeness to P”. 

The question now arises: given two patches whicfc are in- 
distinguishable in colour, what makes me think them two? The 
answer is obvious: difference of spatio-temporal position. But 


HUMAN knowledge: its scope and limits 

though this answer is obvious, it does not dispose of the problem. 
For the sake of simplicity, let us suppose that the two patches are 
parts of one visual field, but are not in visual contact with each 
other. Spatial position in the momentary visual field is a quality, 
varying according to distance from the centre of the field of 
vision, and also according as the region in question is above or 
below, to the right or to the left, of the centre. The various 
qualities that small portions of the visual field may have are 
related by relations of up-and-down, right-and-left. When we 
move our eyes, the qualities associated with a given physical 
object change, but if the various physical objects have not moved, 
there will be no topological change in the part of the visual field 
which is common to both occasions. This enables common sense 
to ignore the subjectivity of visual position. 

Concerning these visual positional qualities we have exactly 
the same alternatives as in the case of shades of colour. We may 
give a name to each quality, considered as something which is the 
same on different occasions, or we may give a name to each 
instance of the quality, and connect it with other instances of the 
same quality by the relation of exact likeness. Let us concentrate 
on the quality that distinguishes the centre of the field of vision, 
and let us call this quality “centrality”. Then on one view there 
is a single quality of centrality, which occurs repeatedly, while 
on the other view there are many particulars which have exact 
positional likeness, and the quality of centrality is replaced by 
the class of these particulars. 

When we now repeat, in relation to the particulars which are 
instances of centrality, the question as to how we distinguish one 
of these particulars from another, the answer is again obvious: 
we distinguish them by their position in time. (There cannot be 
two simultaneous instances of centrality in one person’s experi- 
ence.) We must therefore now proceed to analyse difference of 
position in time. 

In regard to time, as in regard to space, we have to distinguish 
objective and subjective time. Objective space is that of the 
physical world, whereas subjective space is that which appears 
in our percepts when we view the world from one place. So 
objective time is that of physics and history, while subjective time 
is that which appears in our momentary view of the world. In my 
present state of mind there are not only percepts, but memories 



and expectations; what I remember I place in the past, what I 
expect I place in the future. But from the impartial stand-point 
of history my memories and expectations are just as much now 
as my percepts. When I remember, something is happening to 
me now which, if I remember correctly, has a certain relation to 
what happened at an earlier time, but what happened then is not 
itself in my mind now. My memories are placed in a time-order, 
just as my visual perceptions are placed in a space-order, by 
intrinsic qualities, which may be called “degrees of remoteness”. 
But however high a degree of remoteness a memory may possess, 
it is still, from the objective historical point of view, an event 
which is happening now. 

I said a moment ago that there could not be two simultaneous 
instances of centrality in one person’s experience, but in a certain 
sense this may be false. If, when my eyes are open, I remember 
some previous visual experience, there will be one instance of 
centrality in my percepts and another in my memory, and these 
are both now in historical time. But they are not both now in the 
time of my present subjective experience. Thus the correct state- 
ment is: two instances of centrality cannot be simultaneous in 
historical time if they arc perceptual parts of one man’s experience, 
and they cannot in any case be simultaneous in the subjective 
time of a single experience composed of percepts and memories 
and expectations. 

There is a certain difficulty in the conception of a time which, 
in a sense, is wholly now y and a space which, in a sense, is wholly 
here. Yet these conceptions seem unavoidable. The whole of my 
psychological space is here from the standpoint of physics, and 
the whole of my psychological time is now from the standpoint 
of history. Like Leibniz’s monads, we mirror the universe, though 
very partially and very inaccurately ; in my momentary mirroring 
there is a mirror-space and a mirror-time, which have a corre- 
spondence, though not an exact one, with the impersonal space 
and time of physics and history. From the objective standpoint, 
the space and time of my present experience are wholly confined 
within a small region of physical space-time. 

We must now return from this digression to the question 
whether we are to assume one quality of centrality 'which can 
exist at various times, or a number of instances of it, each of which 
exists only once. It begins to be obvious that the latter hypothesis 


HUMAN knowledge: its scope and limits 

will entail great unnecessary complications, which the former 
hypothesis avoids. We can bring the question to a head by asking 
what can be meant by “this”. Let us suppose “this” to be some 
momentary visual datum. There is a sense in which it may be true 
to say “I have seen this before”, and there is another sense in 
which this cannot be true. If I mean by “this” a certain shade of 
colour, or even a certain shade of a certain shape, I may have 
seen it before. But if I mean something dated, such as might be 
called an “event”, then clearly I cannot have seen it before. Just 
the same considerations apply if I am asked “do you see this any- 
where else”? I may be seeing the same shade of colour somewhere 
else, but if in the meaning of “this” I include position in visual 
space, then I cannot be seeing it somewhere else. Thus what we 
have to consider is spatio-temporal particularity. 

If we take the view — which I think the better one — that a 
given quality, such as a shade of colour, may exist in different 
places and times, then what would otherwise be instances of the 
quality become complexes in which it is combined with other 
qualities. A shade of colour combined with a given positional 
quality cannot exist in two parts of one visual field, because the 
parts of the field are defined by their positional qualities. There is 
a similar distinction in subjective time: the complex consisting 
of a shade of colour together with one degree of remoteness 
cannot be identical with the complex consisting of the same shade 
of colour and another degree of remoteness. In this way “instances” 
can be replaced by complexes, and by this replacement a great 
simplification can be effected. 

It results from the above discussion that a possible minimum 
vocabulary for describing the world of my experience can be 
constructed as follows. Names are given to all the qualities of 
experiences, including such qualities of visual space and remem- 
bered time as we have been considering. We also have to have 
words for experienced relations, such as right-and-left in one 
visual field, and earlier-and-later in one specious present. We do 
not need names for space-time regions, such as “Socrates” or 
“France”, because every space-time region can be defined as a 
complex of qualities or a system of such complexes. “Events”, 
which have dates and cannot recur, are capable of being regarded 
as always complex; whatever we do not know how to analyse is 
capable of occurring repeatedly in various parts of space-time. 



When we pass outside our own experience, as we do in physics, 
we need no new words. Definitions of things not experienced 
must be denotational. Qualities and relations, if not experienced, 
can only be known by means of descriptions in which all the 
constants denote things that are experienced. It follows that a 
minimum vocabulary for what we experience is a minimum 
vocabulary for all our knowledge. That this must be the case is 
obvious from a consideration of the process of ostensive definition. 


Chapter V 


T he purpose of this Part is to provide possible interpre- 
tations of the concepts of science, in terms of possible 
minimum vocabularies. It will not be asserted that no other 
interpretations are possible, but it is hoped that, in the course 
of the discussion, certain common characteristics of all acceptable 
interpretations will emerge. In the present chapter we shall be 
concerned to interpret the word “time”. 

Most people will be inclined to agree with St. Augustine: 
“What, then, is time? If no one asks of me, I know: if I wish to 
explain to him who asks, I know not.” Philosophers, of course, 
have learned to be glib about time, but the rest of mankind, 
although the subject feels familiar, are apt to be aware that a few 
questions can reduce them to hopeless confusion. “Does the 
past exist? No. Does the future exist? No. Then only the present 
exists? Yes. But within the present there is no lapse of time? 
Quite so. Then time does not exist ? Oh I wish you wouldn’t be 
so tiresome.” Any philosopher can elicit this dialogue by a suitable 
choice of interlocuter. 

Sir Isaac Newton, who understood the Book of Daniel, also 
knew all about time. Let us hear what he has to say on the subject 
in the Scholium following the initial definitions in the Principia: 

“I do not define time, space, place and motion, as being well 
known to all. Only I must observe, that the vulgar conceive 
those quantities under no other notions but from the relation 
they bear to sensible objects. And thence arise certain prejudices, 
for the removal of which, it will be convenient to distinguish 
them into absolute and relative, true and apparent, mathematical 
and common. Absolute, true, and mathematical time, of itself, 
and from its own nature flows equably without regard to anything 
external, and by another name is called duration : relative, appar- 
ent, and common time, is some sensible external (whether accurate 
or unequable) measure of duration by the means of motion, 
which is commonly used instead of true time; such as an hour, 
a day, a month, a year.” 

He goes on to explain that days are not all of equal length, and 



that perhaps there is nowhere in nature a truly uniform motion, 
but that we arrive at absolute time, in astronomy, by correction 
of “vulgar” time. 

Sir Isaac Newton's “absolute” time, although it remained 
embedded in the technique of classical physics, was not generally 
accepted. The theory of relativity has provided reasons, within 
physics, for its rejection, though these reasons leave open the 
possibility of absolute space-time. But before relativity Newton's 
absolute time was already widely repudiated, though for reasons 
which had nothing to do with physics. Whether, before rela- 
tivity, these reasons had any validity is a question which I think 
we shall find it worth while to examine. 

Although Newton says that he is not going to define time 
because it is well known, he makes it clear that only “vulgar” 
time is well known, and that mathematical time is an inference. 
In modern terms, we should rather call it an adjustment than an 
inference. The process of arriving at “mathematical” time is 
essentially as follows: there are a number of periodic motions — 
the rotations and revolutions of the earth and the planets, the 
tides, the vibrations of a tuning fork, the heart beats of a healthy 
man at rest — which are such that, if one of them is assumed to be 
uniform, all the others are approximately uniform. If we take one 
of them, say the earth's rotation, as uniform by definition, we 
can arrive at physical laws — notably the law of gravitation — 
which explain the phenomena, and show why the other periodic 
motions are approximately uniform. But unfortunately the laws 
so established are only approximate, and, what is more, they show 
that the earth’s rotation should suffer retardation by tidal friction. 
This is self-contradictory if the earth's rotation is taken as the 
measure of time; we therefore seek a different measure, which 
shall also make our physical laws approximate more nearly to 
exact truth. It is found convenient not to take any actual motion 
as defining the measure of time, but to adopt a compromise 
measure which makes physical laws as accurate as possible. 
It is this compromise measure that serves the purposes for which 
Newton invoked “absolute” time. There is no reason, however, 
to suppose that it represents a physical reality, for the choice of 
a measure of time is conventional, like the choice between the 
Christian and the Mohammedan eras. We choose, in fact, the 
measure which gives the greatest attainable simplicity to the 


human knowledge: its scope and limits 

statement of physical laws, but we do so on grounds of conveni- 
ence, not because we think that this measure is more “true” than 
any other. 

A frequent ground of objection to Newton’s “absolute” time 
has been that it could not be observed. This objection, on the 
face of it, comes oddly from men who ask us to believe in electrons 
and protons and neutrons, quantum transitions in atoms, and 
what not, none of which can be observed. I do not think that 
physics can dispense with inferences that go beyond observation. 
The fact that absolute time cannot be observed is not, by itself, 
fatal to the view that it should be accepted; what is fatal is the 
fact that physics can be interpreted without assuming it. When- 
ever a body of symbolic propositions which there is reason to 
accept can be interpreted without inferring such-and-such 
unobserved entities, the inference from the body of propositions 
in question to these supposed entities is invalid, since, even if 
there are no such entities, the body of propositions may be true. 
It is on this ground, and not merely because “absolute” time 
cannot be observed, that Newton was mistaken in inferring it 
from the laws of physics. 

While the rejection of Newton’s view is a commonplace, few 
people seem to realize the problems that it raises. In physics 
there is an independent variable /, the values of which are 
supposed to form a continuous series, and each to be what is 
commonly called an “instant”. Newton regarded an instant as a 
physical reality, but the modern physicist does not. Since, how- 
ever, he continues to use the variable t , he must find some 
interpretation for its values, and the interpretation must serve 
the technical purposes that were served by Newton’s “absolute” 
time. This problem of the interpretation of is the one that 
concerns us in this chapter. In order to simplify the approach to 
it, we will at first ignore relativity and confine ourselves to time 
as it appears in classical physics. 

We shall continue to give the name “instant” to a value of 
the variable t 9 but we shall look for an interpretation of the word 
“instant” in terms of physical data, that is to say, we shall expect 
the word to have a definition, and not to belong to a minimum 
physical vocabulary. All that we require of the definition is that 
instants, so defined, should have the formal properties demanded 
of them by mathematical physics. 



In seeking a definition of “instant” or “point”, the material 
to be used depends upon the theory we adopt as to “particulars” 
or proper names. We may take the view that when, for instance, 
a given shade of colour appears in two separated locations, there 
are two separate “particulars”, each of which is an “instance” 
of the shade of colour, and is a subject of which qualities can 
be predicated, but which is not defined by its qualities, since 
another precisely similar particular might exist elsewhere. Or 
we may take the view that a “particular” is a bundle of coexisting 
qualities. The discussions of the preceding chapter, as well as 
the earlier discussion of proper names, inclined us to the latter 
view. I shall, however, in this and the two following chapters, 
hypothetically adopt the former view, and in Chapter VIII I 
shall show how to interpret what has been said in terms of the 
latter view. For the moment, therefore, I take as raw material 
“events”, which are to be imagined as each occupying a finite 
continuous portion of space-time. It is assumed that two events 
can overlap, and that no event recurs. 

It is clear that time is concerned with the relation of earlier 
and later; it is generally held also that nothing of which we 
have experience has a merely instantaneous existence. Whatever 
is earlier or later than something else I shall call an “event”. 
We shall want our definition of “instant” to be such that an 
event can be said to exist “at” certain instants and not at certain 
others. Since we have agreed that events, so far as known to us, 
are not merely instantaneous, we shall wish to define “instant” 
in such a way that every event exists at a continuous stretch of 
the series of instants. That instants must form a series defined 
by means of the relation of earlier and later is one of the requisites 
that our definition must fulfil. Since we have rejected Newton's 
theory, we must not regard instants as something independent 
of events, which can be occupied by events as hats occupy hat- 
pegs. We are thus compelled to search for a definition which 
makes an instant a structure composed of a suitable selection of 
events. Every event will be a member of many such structures, 
which will be the instants during which it exists: it is “at” every 
instant which is a structure of which the event is a member. 

A date is fixed with complete precision if it is known concerning 
every event in the world whether it wholly preceded that date, or 
will wholly come after it, or was in existence at that date. To this 



statement some one might object that, if the world were to remain 
without change for (say) five minutes, there would be no way 
of fixing a date within these five minutes if the above view were 
adopted, for every event wholly preceding one part of the five 
minutes would wholly precede every other part, every event 
wholly subsequent to any part of the five minutes would be 
wholly subsequent to every other part, and every event existing 
at any part of the five minutes would exist throughout the whole 
of them. This, however, is not an objection to our statement, 
but only to the supposition that time could go on in an unchanging 
world. On the Newtonian theory this would be possible, but on 
a relational theory of time it becomes self-contradictory. If 
time is to be defined in terms of events, it must be impossible for 
the universe to be unchanging for more than an instant. And 
when I say “impossible” I mean logically impossible. 

Although we cannot agree with Newton that “time” does not 
need to be defined, it is obvious that temporal statements demand 
some undefined term. I choose the relation of earlier-and- 
later, or of wholly-preceding. Between two events a and b three 
temporal relations are possible: a may be wholly before b , or 
b may be wholly before a , or a and b may overlap. Suppose 
you wish to fix as accurately as possible some date within the 
duration of a. If you say that your date is also to be within the 
duration of b , you fix the date somewhat more accurately than 
by merely saying that it is within the duration, of a , unless it 
so happens that a and b both began and ended together. Suppose 
now there is a third event c which overlaps with both a and b — 
that is to say, in ordinary language (to which we are not yet 
entitled), there is a period of time during which a and b and c 
all exist. This period, in general, will be shorter than that during 
which both a and b exist. We now look for a fourth event d which 
overlaps with a and b and c y i.e., in ordinary language, exists 
during some part of the time during which a and b and c all 
exist; the time during which a and b and c and d all exist is, 
in general, shorter than that during which any three of them 
all exist. In this way, step by step, we get nearer to an exact date. 

Let us suppose this process carried on as long as possible, i.e. 
until there is no event remaining which overlaps with all the 
events already in our group. I say that, when this stage has been 
reached, the group of events that has been constructed may 



be defined as an “instant”. To prove that this assertion is legiti- 
mate I only have to show that “instants”, so defined, have the 
mathematical properties that physics demands. I do not have 
to show that this is what people commonly mean when they speak 
of “instants”, though it might be desirable to complete the argu- 
ment by showing that they commonly mean nothing. 

An “instant”, as I propose to define the term, is a class of 
events having the following two properties: (i) all the events 
in the class overlap; (2) no event outside the class overlaps with 
every member of the class. This group of events, as I shall show, 
does not persist for a finite time. 

To sav that an event persists for a finite time can only mean, 
on a relational view of time, that changes occur while it exists, 
i.e. that the events which exist when it begins are not all identical 
with the events existing when it ends. This amounts to saying 
that there are events which overlap with the given event but 
not with each other. That is to say: “0 lasts for a finite time” 
means “there are two events b and c such that each overlaps 
with a but b wholly precedes c\ 

We may apply the same definition to a group of events. If 
the members of the group do not all overlap, the group as a whole 
has no duration, but if they all overlap, we shall say that the 
group as a whole lasts for a finite time if there are at least two 
events which overlap with every member of the group although 
one of them wholly precedes the other. If this is the case, change 
occurs while the group persists; if not, not. Now if a group 
constitutes an “instant” as above defined, no event outside the 
group overlaps all the members of the group, and no event inside 
the group wholly precedes any other event inside the group. 
Therefore the group as a whole does not last for a finite time. 
And therefore it may suitably be defined as an “instant”. 

Instants will form a series ordered by a relation defined in 
terms of the relation “wholly preceding” among events. One 
instant is earlier than another if there is a member of the first 
instant which wholly precedes a member of the second, i.e. if 
some event “at” the first instant wholly precedes some event 
“at” the second instant. It will be observed that being “at” an 
instant is the same thing as being a member of the class which is 
the instant. 

According to the above definition, it is logically impossible 

289 T 

HUMAN knowledge: its scope and limits 

for the world to remain unchanging throughout a finite time. If 
two instants differ, they are composed (at least in part) of different 
members, and that means that some event existing at the one 
instant does not exist at the other. 

Our theory makes no assumption as to whether there are or 
are not events that exist only at one instant. Such events, if any, 
would have the characteristic that any two events overlapping 
with them would overlap with each other. In general, the “dura- 
tion” of an event means “the class of those instants of which the 
event in question is a member”. It is generally assumed that an 
event occupies a continuous stretch of the series of instants; 
this assumption, formally, is embodied in the “axiom” that 
nothing wholly precedes itself. But this axiom is not necessary. 

Something has already been said concerning the quantitative 
measurement of time, but it may be well to re-state the view to 
which we are led by physics. The quantitative measurement of 
time is conventional except to this extent, that a larger measure 
must be applied to a whole than to a part. We must assign a 
larger measure to a year than to any month in that year, but we 
might, if convenient, assign to that year a smaller measure than 
to a month in some other year. It turns out, however, that this 
is not convenient. Historically, astronomers started with the 
assumption that the day and the year were each of constant 
length ; then it turned out that, if the sidereal day was constant, 
the solar day was not, but the year was. If the sidereal day was 
constant by definition, a large number of other periodic occur- 
rences were approximately constant; this led to dynamical laws 
which suggested that it would be more convenient to treat the 
sidereal day as not exactly constant owing to tidal friction. The 
laws could be formulated with any measure of time, but naturally 
astronomers and physicists preferred the measure which made 
the statement of the laws simplest. As this so very nearly agreed 
with the “natural” measures of days and years, its conventional 
character was not perceived, and it could be supposed that 
what was being defined was Newton’s “true” or “mathematical” 
time, believed to have physical reality. 

I have been speaking so far as if there were, as used to be 
thought, one cosmic time for the whole universe. Since Einstein, 
we know that this is not the case. Each piece of matter has its 
own local time. There is very little difference between the local 



time of one piece of matter and that of another unless their 
relative velocity is an appreciable fraction of the velocity of light. 
The local time of a given piece of matter is that which will be 
shown by a perfectly accurate chronometer which travels with it. 
Beta-particles travel with velocities that do not fall very far 
short of that of light. If we could place a chronometer on a beta- 
particle, and make the particle travel in a closed path, we should 
find, when it returned, that the chronometer would not agree 
with one that had remained throughout stationary in the 
laboratory. A more curious illustration (which I owe to Professor 
Reichenbach) is connected with the possibility of travel to the 
stars. Suppose we invented a rocket apparatus which could 
send a projectile to Sirius with a velocity ten elevenths of that of 
light. From the point of view of the terrestrial observer the 
journey would take about 55 years, and one might therefore 
suppose that if the projectile carried passengers who were young 
when they started, they would be old when they arrived. But 
from their point of view the journey will only have taken about 
11 years. This will not only be the time taken as measured by 
their clocks, but also the time as measured by their physiological 
processes — decay of teeth, loss of hair, etc. If they looked and 
felt like men of 20 when they started, they will look and feel like 
men of 31 when they arrive. It is only because we do not habitually 
come across bodies travelling with a speed approaching that of 
light that such odd facts remain unnoticed except by men of 

If two pieces of matter (say the earth and a comet) meet and 
part and meet again, and if in the interval their relative velocity 
has been very great, the physicists (if any) who live on the two 
pieces of matter will form different estimates of the lapse of 
time between the two meetings, but they will agree as to which 
of the two meetings was the earlier and which the later. “Earlier” 
and “later”, therefore, as applied to two events happening to one 
piece of matter, have no ambiguity: if there are several pieces 
of matter to which both the given events happen, one of the 
events will be earlier for all of them, and the other will be later 
for all of them. 

The construction of “instants” as classes of events, given 
above, is to be held, for the present, as applying only to events 
happening to one piece of matter — primarily the body of a given 


HUMAN knowledge: its scope and limits 

observer. The extension to cosmic time, which can be made in 
many ways, all equally legitimate, is a matter which I shall not 
deal with at present. 

Instead of basing our construction on the events happening 
to a given body, we may base it on those happening to a given 
mind or forming part of a given experience. If the mind is mine, 
I can experience occurrences of the sort expressed by the words 
“A wholly precedes B”, for example, when I am listening to 
successive strokes of a clock striking the hour. If A is an event 
that I experience, everything that overlaps with A or wholly 
precedes A or wholly succeeds A will constitute “my” time, and 
only events belonging to “my” time will be involved in the 
construction of “instants” belonging to “my” time . 1 The linking 
up of my time with yours will thus remain a problem to be con- 

We may define a “biography” as a collection of events such 
that, of any two, either they overlap or there is one that wholly 
precedes the other. For the present I shall assume that, when a 
biography has a psychological definition, it also has a physical 
definition — i.e. the time-series constituted out of events that I 
experience is identical with the time-series constructed out of 
events that happen to my brain, or some part of it. Accordingly, 
I shall speak of the “biography” of a piece of matter, not only 
of the “biography” connected with some person’s experience. 

What has been said so far can now be summed up in a series 
of definitions. 

An “event” is something which precedes or follows or overlaps 

The “biography” to which an event belongs is all the events 
that it precedes or follows or overlaps. 

An “instant” is a collection of events belonging to one bio- 
graphy, and having the two properties that (a) any two events in 
the collection overlap, ( b ) no event outside the collection over- 
laps with all the members of the collection. 

An event is said to “exist at” an instant if it is a member of 
that instant. 

One instant is said to be “earlier” than another if there is an 
event at the one which wholly precedes some event at the other. 

i “My” time in the above sense is not to be confused with the sub- 
jective time of Part III, Chapter V. 



A “time-series of a given instant” is a series of instants, of 
which the given instant is one, and having the property that, 
of any two, one is earlier than the other. 

A “time-series” is a time-series of some instant. 

It is not assumed that an instant can only belong to one time- 
series, nor that an event can only belong to one biography. But 
it is assumed that, if a wholly precedes 6, then a and b are not 
identical. This is an assumption which we shall have to examine, 
and perhaps modify, at a later stage. 

As the above construction of time-series is the simplest example 
of a kind of procedure which will be frequently employed, I 
shall spend a few moments in setting forth the reasons for its 

We start from the fact that, although physicists reject Newton’s 
absolute time, they continue to employ the independent variable 
t, of which the values are said to be “instants”. The values of t 
are held to form a series ordered by a relation called “earlier- 
and-later”. It is held also that there are occurrences called 
“events”, which include as a sub-class everything that we can 
observe. There are two observable temporal relations among 
events: they may overlap, as when I hear a clock striking while 
I see its hands pointing to twelve o’clock; or one may precede 
another, as when I still remember the previous stroke of the 
clock while I am hearing the present stroke. These are the data 
of our problem. 

Now if we are to use the variable t without assuming Newton’s 
absolute time, we must find a way of defining the class of values 
of t 9 that is to say, “instants” must not form part of our minimum 
vocabulary, which, so far as it is not merely that of logic, must 
consist of words whose meaning is known by experience. 

Definitions are of two sorts, which may be called respectively 
“denotational” and “structural”. An example of a denotational 
definition is “the tallest man in the United States”. This is 
certainly a definition, since there must be one and only one 
person to whom it applies, but it defines the man merely by his 
relations. Generally, a “denotational” definition is one which 
defines an entity as the only one having a certain relation, or 
certain relations, to one or more known entities. On 'the other 
hand, when what we want to define is a structure composed of 
known elements, we can define it by mentioning the elements and 


HUMAN knowledge: its scope and limits 

the relations constituting the structure; this is what I call a 
“structural” definition. If what I am defining is a class, it may 
be only necessary to mention the structure, since the elements 
may be irrelevant. For example, I can define an “octagon” as 
“a plane figure having eight sides”; this is a structural definition. 
But I might also define it as “a polygon of which all known 
examples are in the following places”, then giving a list. This 
would be a “denotational” definition. 

A denotational definition is not complete without a proof of 
the existence of the object denoted. “The man over io feet 
tall” is, in logical form, just as good as “the tallest man in the 
United States”, but it probably denotes no one. “The square 
root of 2” is a denotational definition, but until our own day 
there was no proof that it denoted anything; now we know 
that it is equivalent to the structural definition “the class of 
those rationals whose squares are less than 2”, and thereby 
the question of “existence” (in the logical sense) is solved. Owing 
to the possible doubt about “existence”, denotational definitions 
are often unsatisfactory. 

In the particular case of our variable t , a denotational definition 
is excluded by our rejection of absolute time. We must therefore 
seek a structural definition. This implies that instants must have 
a structure, and that the structure must be built out of known 
elements. We have, as data of experience, the relations “over- 
lapping” and “preceding”, and we find that by means of these 
we can build structures having the formal properties that mathe- 
matical physicists demand of “instants”. Such structures, there- 
fore, fulfil all required purposes without the need of any ad hoc 
assumption. This is the justification of our definitions. 


Chapter VI 


I N this chapter we shall be concerned with space as it appears 
in classical physics. That is to say, we shall be concerned to 
find an “interpretation” (not necessarily the only possible one) 
for the geometrical terms used in physics. Much more complicated 
and difficult problems arise in regard to space than in regard to 
time. This is partly because of problems introduced by relativity. 
For the present, however, we will ignore relativity, and treat 
space as separable from time after the manner of pre-Einsteinian 

For Newton, space, like time, was “absolute”, that is to say, 
it consisted of a collection of points, each devoid of structure, 
and each one of the ultimate constituents of the physical world. 
Each point was everlasting and unchanging; change consisted in 
its being “occupied” sometimes by one piece of matter, sometimes 
by another, and sometimes by nothing. As against this view, 
Leibniz contended that space was only a system of relations, 
the terms of the relations being material and not merely geo- 
metrical points. Although both physicists and philosophers 
tended more and more to take Leibniz's view rather than 
Newton’s, the technique of mathematical physics continued to 
be Newtonian. In the mathematical apparatus, “space” is still 
an assemblage of “points”, each defined by three co-ordinates, 
and “matter” is an assemblage of “particles”, each of which 
occupies different points at different times. If we are not to 
agree with Newton in ascribing physical reality to points, this 
system requires some interpretation in which “points” have a 
structural definition. 

I have used the word “physical reality”, which may be held 
to savour too much of metaphysics. What I mean can be expressed, 
in a form more acceptable to modern taste, by means of the 
technique of minimum vocabularies. Given a collection of names, 
it may happen that some of the things named have a structural 
definition in terms of others; in that case, there will be & minimum 
vocabulary not containing the names for which definitions can be 
substituted. For example, every French human being has a 


HUMAN knowledge: its scope and limits 

proper name, and “the French nation” may also be regarded as 
a proper name, but it is an unnecessary one, since we can say: 
“the French nation” is defined as “the class consisting of the 
following individuals (here follows the list)”. Such a method is 
only applicable to finite classes, but there are other methods not 
subject to this limitation. We can define “France” by its geo- 
graphical boundaries, and then define “French” as “born in 

To this process of substituting structural definitions for names 
there are obviously limits in practice, and perhaps (though 
this may be questioned) there are also limits in theory. Assuming, 
for the sake of simplicity, that matter consists of electrons and 
protons, we could, in theory, give a proper name to each electron 
and each proton; we could then define an individual human 
being by mentioning the electrons and protons composing his 
body at various times; thus names for individual human beings 
are theoretically superfluous. Speaking generally, whatever has a 
discoverable structure does not need a name, since it can be 
defined in terms of the names of its ingredients and the words 
for their relations. On the other hand, whatever has no known 
structure needs a name if we are to be able to express all our 
knowledge concerning it. 

It is to be observed that a denotational definition does not 
make a name superfluous. E.g. “the father of Alexander the 
Great” is a denotational definition, but does not enable us to 
express the fact which contemporaries could have expressed 
by “that is Alexander’s father”, where “that” functions as a 

When we deny Newton’s theory of absolute space, while 
continuing to use what we call “points” in mathematical physics, 
our procedure is only justified if there is a structural definition 
of “point” and (in theory) of particular points. Such a definition 
must proceed by methods similar to those that we employed in 
defining “instants”. This, however, is subject to two provisos: 
first, that our manifold of points is to be three-dimensional, and 
second, that we have to define a point at an instant. To say that 
a point P at one time is identical with a point Q at another time 
is to say something which has no definite meaning except a 
conventional one which depends upon a choice of material axes. 
As this matter has to do with relativity, I shall not consider it 



further at present, but shall confine myself to the definition of 
points at a given instant, ignoring the difficulties connected with 
the definition of simultaneity. 

In what follows I lay no stress on the particular method of 
constructing points that I have adopted. Other methods are 
possible, and some of these may be preferred. What is important 
is only that such methods can be devised. In defining instants, 
we used the relation of “overlapping” in a temporal sense — a 
relation which holds between two events when (in ordinary 
language) there is a time during which both exist. In defining 
points, we use the relation of “overlapping” in a spatial sense, 
which is to subsist between two simultaneous events that (in 
ordinary language) occupy the same region of space, in whole 
or in part. It is to be observed that events, unlike pieces of matter, 
are not to be thought of as mutually impenetrable. The impene- 
trability of matter is a property which results tautologically from 
its definition. “Events”, however, are only defined as terms not 
assumed to possess a structure, and having spatial and temporal 
relations such as belong to finite volumes and finite periods of 
time. When I say “such as”, I mean “similar as regards logical 
properties”. But “overlapping” is not itself to be defined logically; 
it is an empirically known relation, having, in the construction 
which I advocate, only an ostensive definition. 

In a manifold of more than one dimension, we cannot construct 
anything having the properties required of “points” by means 
of a two-term relation of “overlapping”. As 
the simplest illustration, let us take areas on ^ I B 

a plane. Three areas A, B, C, on a plane may A t>Jr 

each overlap with the other two, without there 

being any region common to all three. In the 
accompanying figure, the circle A overlaps with the rectangle B 
and the triangle C, and B overlaps with C, but there is no region 
common to A and B and C. The basis of our construction will 
have to be a relation of three areas, not of two. We shall say that 
three areas are “copunctual” when there is a region common to all 
three. (This is an explanation, not a definition.) 

We shall assume that the areas with which we are concerned 
are all either circles, or such shapes as can result from circles 
by stretching or compressing in a manner which leaves them oval. 
In that case, given three areas A, B, C which are copunctual, and 


a fourth area D such that A, B, D are copunctual and also A, C, D 
and B, C, D, then A, B, C, D all have a common region. 

We now define a group of any number of areas as “copunctual” 
if every triad chosen out of the group is copunctual. A copunctual 
group of areas is a “point” if it cannot be enlarged without 
ceasing to be copunctual, i.e. if, given any area X outside the 
group, there are in the group at least two areas A and B such that 
A, B, X are not copunctual. 

This definition is only applicable in two dimensions. In three 
dimensions, we must start with a relation of copunctuality between 
four volumes, and the volumes concerned must all be either 
spheres or such oval volumes as can result from spheres by 
continuous stretching in some directions and compressing in 
others. Then, as before, a copunctual group of volumes is one 
in which every four are copunctual, and a copunctual group is 
a “point” if it cannot be enlarged without ceasing to be co- 

In n dimensions the definitions are the same, except that the 
original relation of copunctuality has to be between n + 1 regions. 

“Points” are defined as classes of events by the above 
methods, with the tacit assumption that every event “occupies” 
a more or less oval region. 

“Events” are to be taken, in the present discussion, as the 
undefined raw material from which geometrical definitions are 
to be derived. In another context we may inquire as to what can 
be meant by an “event”, and we may then be able to carry 
analysis a step further, 1 but for the present we regard the manifold 
of “events”, with their spatial and temporal relations, as empirical 

The way in which spatial order results from our assumptions 
is somewhat complicated. I shall say nothing about it here, as 
I have dealt with it in Analysis of Matter , where, also, there is 
a much fuller discussion of the definition of “points” (Chapters 

Something must be said about the metrical properties of space. 
Astronomers, in their popular books, astound us first by telling 
us how immensely distant many of the nebulae are, and then by 
telling us that after all the universe is finite, being a three- 
dimensional analogue of the surface of a sphere. But in their less 
1 See Part II, Chapter III and Part IV, Chapter IV. 


popular books they tell us that measurement is merely conven- 
tional, and that we could, if we chose, adopt a convention which 
would make the furthest known nebula in the northern hemisphere 
nearer to us than the antipodes are. If so, the vastness of the 
universe is not a fact, but a convenience. I think this is only 
partially true, but to disentangle the element of convention in 
measurement is by no means easy. Before attempting it, some- 
thing must be said about measurement in its elementary forms. 

Measurement, even of the distance to remote nebulae, is built 
up from measurements of distances on the surface of the earth, 
and terrestrial measurements start with the assumption that certain 
bodies may be regarded as approximately rigid. If you measure 
the size of your room, you assume that your foot-rule is not 
growing appreciably longer or shorter during the process. The 
ordnance survey of England determines most distances by tri- 
angulation, but this process demands that there shall be at least 
one distance which is measured directly. In fact, a base line on 
Salisbury Plain was chosen, and was measured carefully in the 
elementary way in which we measure the size of our room: a 
chain, which we may take as by definition of unit length, was 
repeatedly applied to the surface of the earth along a line as nearly 
straight as possible. This one length having been determined 
directly, the rest proceeds by the measurement of angles and by 
calculation: the diameter of the earth, the distance of the sun 
and moon, and even the distances of the nearer fixed stars, can 
be determined without any further direct measurement of lengths. 

But when this process is scrutinized it is found to be full of 
difficulties. The assumption that a body is “rigid” has no clear 
meaning until we have already established a metric enabling us 
to compare lengths and angles at one time with lengths and angles 
at another, for a “rigid” body is one which does not alter its shape 
or size. Then again we need a definition of a “straight line”, for 
all our results will be wrong if the base line on Salisbury Plain 
and the lines used in triangulation are not straight. It seems, 
therefore, that measurement presupposes geometry (to enable us 
to define “straight lines”) and enough physics to give grounds 
for regarding some bodies as approximately rigid, and for com- 
paring distances at one time with distances at another. The 
difficulties involved are formidable, but are concealed by assump- 
tions taken over from common sense. 


HUMAN knowledge: its scope and limits 

Common sense assumes, roughly speaking, that a body is rigid 
if it looks rigid. Eels do not look rigid, but steel bars do. On the 
other hand, a pebble at the bottom of a rippling brook may look 
as wriggly as an eel, but common sense nevertheless holds it 
to be rigid, because common sense regards the sense of touch 
as more reliable than the sense of sight, and if you wade across 
the brook in bare feet the pebble feels rigid. Common sense, in 
so thinking, is Newtonian: it is convinced that at each moment 
a body intrinsically has a certain shape and size, which either are 
or are not the same as its shape and size at another moment. 
Given absolute space, this conviction has a meaning, but without 
absolute space it is prima facie meaningless. There must, however, 
be an interpretation of physics which will account for the very 
considerable measure of success resulting from common-sense 

As in the case of the measurement of time, three factors enter 
in: first, an assumption liable to correction; second, physical laws 
which, on this assumption, are found to be approximately true; 
third, a modification of the assumption to make the physical laws 
more nearly exact. If you assume that a certain steel rod, which 
looks and feels rigid, preserves its length unchanged, you will find 
that the distance from London to Edinburgh, the diameter of the 
earth, and the distance of Sirius, are all nearly constant, but are 
slightly less in warm weather than in cold. It will then occur to 
you that it will be simpler to say that your steel rod expands with 
heat, particularly when you find that this enables you to regard 
the above distances as almost exactly constant, and, further, that 
you can see the mercury in the thermometer taking up more space 
in warm weather. You therefore assume that apparently rigid 
bodies expand with heat, and you do so in order to simplify the 
statement of physical laws. 

Let us get clear as to what is conventional and what is physical 
fact in this process. It is a physical fact that if two steel rods, 
neither of which feels either hot or cold, look as if they were 
of the same length, and if, then, you heat one by the fire and 
put the other in snow, when you first compare them again the 
one that has been by the fire looks slightly longer than the one 
that has been in the snow, but when both have again reached the 
temperature of your room this difference will have vanished. I 
am here assuming pre-scientific methods of estimating tempera- 



ture: a hot body is one that feels hot, and a cold body is one that 
feels cold. As a result of such rough pre-scientific observations 
we decide that the thermometer gives an exact measure of some- 
thing which is measured approximately by our feelings of heat 
and cold; we can then, as physicists, ignore these feelings and 
concentrate on the thermometer. It is then a tautology that my 
thermometer rises with an increase of temperature, but it is a 
substantial fact that all other thermometers likewise do so. This 
fact states a similarity between the behaviour of my thermometer 
and that of other bodies. 

But the element of convention is not quite as I have just stated 
it. I do not assume that my thermometer is right by definition; 
on the contrary, it is universally agreed that every actual ther- 
mometer is more or less inaccurate. The ideal thermometer, to 
which actual thermometers only approximate, is one which, if 
taken as accurate, makes the general law of the expansion of bodies 
with rising temperature as exactly true as possible. It is an 
empirical fact that, by observing certain rules in making ther- 
mometers, we can make them approximate more and more closely 
to the ideal thermometer, and it is this fact which justifies the 
conception of temperature as a quantity having, for a given body 
at a given time, some exact value which is likely to be slightly 
different from that shown by any actual thermometer. 

The process is the same in all physical measurements. Rough 
measurements lead to an approximate law ; changes in the measur- 
ing instruments (subject to the rule that all instruments for 
measuring the same quantity must give as nearly as possible the 
same result) are found capable of making the law more nearly 
exact. The best instrument is held to be the one that makes the 
law most nearly exact, and it is assumed that an ideal instrument 
would make the law quite exact. 

This statement, though it may seem complicated, is still not 
complicated enough. There is seldom only one law involved, and 
very often the law itself is only approximate. Measurements of 
different quantities are interdependent, as we have just seen in 
the case of length and temperature, so that a change in the way 
of measuring one quantity may alter the measure of another. 
Laws, conventions, and observations are almost inextricably 
intertwined in the actual procedure of science. The result of an 
observation is usually stated in a form which assumes certain laws 


HUMAN knowledge: its scope and limits 

and certain conventions; if the result contradicts the network of 
laws and conventions hitherto assumed, there may be considerable 
liberty of choice as to which should be modified. The stock 
instance is the Michelson-Morley experiment, where the simplest 
interpretation was found to involve a radical change in temporal 
and spatial measurements. 

Let us now return to the measurement of distance. There are 
a number of rough pre- scientific observations which suggest the 
methods of measurement actually adopted. If you walk or bicycle 
along a level road with what feels like constant exertion, you will 
take roughly equal times for successive miles. If the road has 
to be tarred, the amount of material required for one mile will 
be about the same as that required for another. If you motor 
along the road, the time taken for each mile will be about what 
your speedometer would lead you to expect. If you base trigono- 
metrical calculations on the assumption that successive miles are 
equal, the results will be in close agreement with those obtained 
directly by measurement. And so on. All this shows that the 
numbers obtained by the usual processes of measurement have 
considerable physical importance, and give a basis for many 
physical and physiological laws. But these laws, once formulated, 
give a basis for amending processes of measurement, and for 
regarding the result of the amended processes as more “accurate”, 
though in fact they are only more convenient. 

There is, however, one element in the notion of “accuracy” 
which is not merely convenient. We are accustomed to the axiom 
that things that are equal to the same thing are equal to one 
another. This axiom has a specious and deceptive appearance 
of obviousness, in spite of the fact that the empirical evidence is 
against it. You may find that, by the most delicate tests you can 
apply, A is equal to B, and B to C, but A is noticeably unequal 
to C. When this happens, we say that A is not really equal to B, 
or B to C. Oddly enough, this tends to be confirmed when the 
technique of measurement is improved. But the real basis of our 
belief in the axiom is not empirical. We believe that equalitv 
consists in possession of a common property. Two lengths arc 
equal if they have the same magnitude , and it is this magnitude 
that we seek to express when we measure. If we are right in this 
belief, the axiom is logically necessary. If A and B have the same 
magnitude, and B and C have the same magnitude, then, of 



necessity, A and C have the same magnitude, provided that 
nothing has more than one magnitude. 

Although this belief in a magnitude, as a property which several 
measurable things may have in common, obscurely influences 
common sense in its conceptions as to what is obvious, it is not 
a belief which we ought to accept until we have evidence of its 
truth in the particular subject-matter concerned. The belief that 
there is such a property of each of a set of terms is logically 
equivalent to the belief that there is a transitive symmetrical 
relation which holds between any two terms of the set. (This 
equivalence is what I formerly called the “principle of abstrac- 
tion’ \) Thus in maintaining that there is a set of magnitudes 
called “distances”, what we are maintaining is this: Between any 
one point-pair and any other, there is either a symmetrical tran- 
sitive relation or an asymmetrical transitive relation. In the former 
case, we say that the distance between the one pair of points is 
equal to the distance between the other pair; in the latter case 
we say that the first distance is less or greater than the second, 
according to the sense of the relation. The distance between two 
points may be defined as the class of point-pairs having to it the 
relation of equidistance. 

This is as far as we can carry the question of the measurement 
of distance without going into the question of the definition of 
straight lines, which we must now consider. 

The straight line has its common-sense origin as an optical 
concept. Some lines look straight. If a straight rod is held, end 
on, against the eye, the part nearest the eye hides all the rest, 
whereas if the rod is crooked some of it will appear round the 
corner. There are of course also other common-sense reasons for 
the concept of straight lines. If a body is rotated, there is a straight 
line, the axis of rotation, which remains unmoved. If you are 
standing up in the Underground, you can tell when it goes round 
a curve by your tendency to overbalance one way or the other. 
It is also possible, up to a point, to judge straightness by the 
sense of touch ; blind men become almost as good at judging shapes 
as men who can see. 

In elementary geometry straight lines are defined as wholes; 
their chief characteristic is that a straight line is determinate as 
soon as two of its points are given. The possibility of regarding 
distance as a straightforward relation between two points depends 


HUMAN knowledge: its scope and limits 

upon the assumption that there are straight lines. But in the 
modern geometry developed to suit the needs of physics there 
are no straight lines in the Euclidean sense, and “distance” is 
only definite when the two points concerned are very close 
together. When the two points are far apart, we must first decide 
by what route we are to trave’ from the one to the other, and 
then add up the many small distances along the route. The 
“straightest” line between the two is the one that makes this 
sum a minimum. Instead of straight lines we have to use “geo- 
desics”, which are routes from one point to another that are shorter 
than any slightly different routes. This destroys the simplicity 
of the measurement of distances, which becomes dependent upon 
physical laws. The resulting complications in the theory of 
geometrical measurement cannot be dealt with without examining 
more closely the connection of physical laws with the geometry 
of physical space. 


Chapter VII 

E verybody knows that Einstein substituted space-time for 
space and time, but people unfamiliar with mathematical 
physics have usually only a very vague conception as to 
the nature of the change. As it is an important change in relation 
to our attempts to conceive the structure of the world, I shall 
try, in this chapter, to explain those parts of it that have philoso- 
phical importance. 

Perhaps the best starting-point is the discovery that “simul- 
taneity” is ambiguous when applied to events in different places. 
Experiments, especially the Michelson-Morley experiment, led 
to the conclusion that the velocity of light is the same for all 
observers, however they may be moving. This seemed, at first 
sight, to be a logical impossibility. If you are in a train which 
is moving at 30 miles an hour, and you are passed by a train 
which is moving at 60 miles an hour, its speed relatively to you 
will be 30 miles an hour. But if it is moving with the velocity 
of light, its speed relatively to you will be the same as its speed 
relatively to fixed points on the earth. Beta particles sometimes 
move with speeds up to 90 per cent of the velocity of light, but 
if a physicist could move with such a particle, and be passed by 
a light-ray, he would still judge that the light was moving, 
relatively to him, at the same rate as if he were at rest in relation 
to the earth. This paradox is explained by the fact that different 
observers, all equipped with perfect chronometers, will form 
different estimates of time-intervals and different judgments as 
to simultaneity in different places. 

It is not difficult to see the necessity for such differences when 
once it has been pointed out. Suppose an astronomer observes 
an event in the sun, and notes the time of his observation; he 
will infer that the event happened about eight minutes before his 
observation, since that is the length of time that it takes light to 
travel from the sun to the earth. But now suppose that the earth 
were travelling very fast towards the sun or away from it. Unless 
you already knew at what moment, by terrestrial time, the event 
on the sun took place, you would not know how far the light had 



HUMAN knowledge: its scope and limits 

had to travel, and therefore your observation would not enable 
you to know when the event in the sun had taken place. That 
is tp say, there would be no definite answer to the question : what 
eft^iits on earth were simultaneous with the solar event that you 
had observed ? 

From the ambiguity of simultaneity it follows that there is a 
parallel ambiguity in the conception of distance. If two bodies 
are in relative motion, their distance apart is continually changing, 
and in pre-relativity physics there was supposed to be such a 
quantity as their “distance at a given instant”. But if there is 
ambiguity as to what is the same instant for the two bodies, there 
is also ambiguity as to “distance at a given instant”. One observer 
will form one estimate, and another will form another, and there 
is no reason to prefer either estimate. In fact, neither time- 
intervals nor spatial distances are facts independent of the move- 
ments of the observer's body. There is a kind of subjectivity 
about measurements of time and space separately — not a psycho- 
logical but a physical subjectivity, since it affects instruments, 
not only mental observers. It is like the subjectivity of the 
camera, which takes a photograph from a certain point of 
view. Photographs from other points of view would look 
different, and no one among them would have a claim to special 

There is, however, one relation between two events which is 
the same for all observers. Formerly there were two, distance in 
space and lapse of time, but now there is only one, which is called 
“interval”. It is because there is only this one relation of interval, 
instead of distance and lapse of time, that we have to substitute 
the one concept of space-time for the two concepts of space and 
time. But although we can no longer separate space and time, 
there are still two kinds of interval, one space-like and the other 
time-like. The interval is space-like if a light-signal, sent out by 
the body on which one event occurs, reaches the body on which 
the other event occurs after this other event has taken place. 
(It is to be observed that there is no ambiguity about the time- 
order of events on a given body.) It is time-like if a light-signal 
sent out from one event reaches the body on which the other 
event occurs before this other event has taken place. Since nothing 
travels faster than light, we may say that the interval is time-like 
when one event may have an effect upon the other, or upon 



something in the same space-time region as the other ; when this 
is not possible, the interval is space-like. 

In the special theory of relativity the definition of “interval*’ 
is simple; in the general theory it is more complicated. 

In the special theory, suppose that an observer, treating himself 
as motionless, judges the distance between two events to be r, 
and the lapse of time between them to be t . Then if c is the 
velocity of light, the square of the interval is 

c 2 1 2 — r 2 

if it is time-like, while if it is space-like it is 

r 2 — c 2 t 2 

It is usually simpler technically to take it as always one of these, 
in which case the square of the other sort of interval is negative, 
and the interval is imaginary. 

When neither gravitation nor electromagnetic forces are in- 
volved, it is found that the interval, as above defined, is the same 
for all observers, and may therefore be regarded as a genuine 
physical relation between the two events. 

The general theory of relativity removes the above restriction 
by introducing a modified definition of “interval”. 

In the general theory of relativity there is no longer a definite 
“interval” between distant events, but only between events that 
are very near together. At a great distance from matter the formula 
for interval approximates to that in the special theory, but else- 
where the formula varies according to the nearness of matter. It 
is found that the formula can be so adjusted as to account for 
gravitation, assuming that matter which is moving freely moves 
in a geodesic, i.e. chooses the shortest or longest route from any 
one point to a neighbouring point. 

It is assumed that, independently of interval, space- time points 
have an order, so that, along any route, one point can be between 
two others which are near it. For example, the interval between 
two different points on one light-ray is zero, but the points still 
have a temporal order: if a ray travels outward from the sun, 
the parts near the sun are earlier than the parts farther from it. 
The space-time order of events is presupposed in the assignment 
of co-ordinates, for although this is to a great extent conventional 
it must always be such that neighbouring points have co-ordinates 
that do not differ much, and that, as points approach closer to 


HUMAN knowledge: its scope and limits 

each other, the difference between their co-ordinates approaches 
zero as a limit. 

If the physical world is held to consist of a four-dimensional 
manifold of events, instead of a manifold of persistent moving 
particles, it becomes necessary to find a way of defining what 
is meant when we say that two events are part of the history of 
one and the same piece of matter. Until we have such a definition, 
“motion” has no definite meaning, since it consists in one thing 
being in different places at different times. We must define a 
“particle”, or material point, as a series of space-time points 
having to each other a causal relation which they do not have 
to other space- time points. There is no difficulty of principle about 
this procedure. Dynamical laws are habitually stated on the 
assumption that there are persistent particles, and are used to 
decide whether two events A and B belong to the biography of 
one particle or not. We merely retain the laws, and turn the 
statement that A and B belong to the same biography into a 
definition of a “biography”, whereas before it seemed to be a 
substantial assertion. 

This point perhaps needs some further explanation. Starting 
from the assumption that there are persistent pieces of matter, 
we arrive at physical laws connecting what happens to a piece 
of matter at one time with what happens to it at another. (The 
most obvious of such laws is the law of inertia.) We now state 
these laws in a different way: we say that, given an event of a 
certain kind in a certain small region of space-time, there will 
be neighbouring events in neighbouring regions which will be 
related to the given event in certain specific ways. We say that 
a series of events related to each other in these specific ways is 
to be called one piece of matter at different times. Thus matter 
and motion cease to be part of the fundamental apparatus of 
physics. What is fundamental is the four-dimensional manifold 
of events, with various kinds of causal relations. There will be 
relations making us regard the events concerned as belonging to 
one piece of matter, others making us regard them as belonging 
to different but interacting pieces of matter, others relating a 
piece of matter to its “empty” environment (e.g. emission of 
light), and yet others relating events that are both in empty space, 
e.g. parts of one light- ray. 

The collecting of events into series such as will secure the 



persistence of matter is only partially and approximately possible. 
When an atom is pictured as a nucleus with planetary electrons, 
we cannot say, after a quantum transition, that such-and-such' 
an electron in the new state is to be identified with such-and-such 
an electron in the old state. We do not even know for certain that 
the number of electrons in the universe is constant. Mass is only 
a form of energy, and there is no reason why matter should not 
be dissolved into other forms of energy. It is energy, not matter, 
that is fundamental in physics. We do not define energy ; we merely 
discover laws as to the changes in its distribution. And these laws 
are no longer such as to determine a unique result where atomic 
phenomena are concerned, though macroscopic occurrences 
remain statistically determinate with an enormously high degree 
of probability. 

The continuity of space-time, which is technically assumed in 
physics, has nothing in its favour except technical convenience. 
It may be that the number of space-time points is finite, and that 
space- time has a granular structure, like a heap of sand. Provided 
the structure is fine enough, there will be no observable phenomenon 
to show that there is not continuity. Theoretically, there might be 
evidence against continuity, but there could never be conclusive 
evidence in its favour. 

The theory of relativity does not affect the space and time of 
perception. My space and time, as known in perception, are cor- 
related with those that, in physics, are appropriate to axes that 
move with my body. Relatively to axes tied to a given piece of 
matter, the old separation of space and time still holds ; it is only 
when we compare two sets of axes in rapid relative motion that 
the problems arise which the theory of relativity solves. Since no 
two human beings have a relative velocity approaching that of 
light, comparison of their experiences will reveal no such dis- 
crepancies as would result if aeroplanes could move as fast as 
beta particles. In the psychological study of space and time, 
therefore, the theory of relativity may be ignored. 


Chapter VIII 


1 shall discuss in this Chapter the modern form of a very 
old problem, much discussed by the scholastics, but still, in 
our day, far from being definitively solved. The problem, in 
its broadest and simplest terms, is this: “How shall we define 
the diversity which makes us count objects as two in a census? 0 
We may put the same problem in words that look different, 
e.g. “what is meant by a ‘particular’?” or “what sort of objects 
can have proper names?” 

Three views have been influentially advocated. 

First: a particular is constituted by qualities; when all its 
qualities have been enumerated, it is fully defined. This is the 
view of Leibniz. 

Second : a particular is defined by its spatio-temporal position. 
This is the view of Thomas Aquinas as regards material substances. 

Third: numerical diversity is ultimate and indefinable. This, 
I think, would be the view of most modern empiricists if they 
took the trouble to have a definite view. 

The second of the above three theories is reducible to either 
the first or the third according to the way in which it is interpreted. 
If we take the Newtonian view, according to which there actually 
are points, then two different points are exactly alike in all their 
qualities, and their diversity must be that bare numerical diversity 
contemplated in the third theory. If, on the other hand, we take 
— as every one now does — a relational view of space, the second 
theory will have to say: “If A and B differ in spatio-temporal 
position, then A and B are two”. But here there are difficulties. 
Suppose A is a shade of colour: it may occur in a number of 
places and yet be only one. Therefore our A and B must not be 
qualities, or, if they are, they must be qualities that never recur. 
If they are not qualities or bundles of qualities, they must be 
particulars of the sort contemplated in our third theory; if they 
are qualities or bundles of qualities, it is the first of our three 
theories that we are adopting. Our second theory, therefore, may 
be ignored. 

The construction of points and instants in our three preceding 



Chapters used “events” as its raw material. Various reasons, of 
which the theory of relativity has been the most influential, have 
made this procedure preferable to one which, like Newton’s, 
allows “points”, “instants”, and “particles” as raw material. It 
has been assumed, in our constructions, that a single event may 
occupy a finite amount of space-time, that two events may overlap 
both in space and in time, and that no event can recur. That is 
to say, if A wholly precedes B, A and B are not identical. We 
assumed also that, if A wholly precedes B, and B wholly pre- 
cedes C, then A wholly precedes C. “Events” were provision- 
ally taken as “particulars” in the sense of our third theory. It was 
shown that, if a raw material of this sort is admitted, space-time 
points and space-time order can be constructed. 

But we are now concerned with the problem of constructing 
space-time points and space-time order when our first theory is 
adopted. Our raw material will now contain nothing that cannot 
recur, for a quality can occur in any number of separate places. 
We have therefore to construct something that does not recur, and 
until we have done so we cannot explain space-time order. 

We have to ask ourselves: what is meant by an “instance”? 
Take some definite shade of colour, which we will call “C”. Let 
us assume that it is a shade of one of the colours of the rainbow, 
so that it occurs wherever there is a rainbow or a solar spectrum. 
On each occasion of its occurrence, we say that there is an 
“instance” of C. Is each instance an unanalysable particular, of 
which C is a quality ? Or is each instance a complex of qualities 
of which C is one? The former is the third of the above theories; 
the latter is the first. 

There are difficulties in either view. Taking first the view that 
an instance of C is an unanalysable particular, we find that we 
encounter all the familiar difficulties connected with the traditional 
notion of “substance”. The particular cannot be defined or recog- 
nized or known ; it is something serving the merely grammatical 
purpose of providing the subject in a subject-predicate sentence 
such as “this is red”. And to allow grammar to dictate our 
metaphysic is now generally recognized to be dangerous. 

It is difficult to see how something so unknowable as such a 
particular would have to be can be required for the interpretation 
of empirical knowledge. The notion of a substance as a peg on 
which to hang predicates is repugnant, but the theory that we 

3 11 


have been considering cannot avoid its objectionable features. I 
conclude, therefore, that we must, if possible, find some other 
way of defining space-time order. 

But when we abandon particulars in the sense which we have 
just decided to reject, we are faced, as observed above, with the 
difficulty of finding something that will not be repeated. A simple 
quality, such as the shade of colour C, cannot be expected to 
occur only once. We shall seek to escape this difficulty by con- 
sidering a “complex” of qualities. What I mean will be most 
easily understood if stated in psychological terms. If I see some- 
thing and at the same time hear something else, my visual and 
auditory experiences have a relation which I call “compresence”. 
If at the same moment I am remembering something that hap- 
pened yesterday and anticipating with dread a forthcoming visit 
to the dentist, my remembering and anticipating are also “corn- 
present” with my seeing and hearing. We can go on to form 
the whole group of my present experiences and of everything 
compresent with all of them. That is to say, given any group 
of experiences which are all compresent, if I can find anything 
else which is compresent with all of them I add it to the group, 
and I go on until there is nothing further which is compresent 
with each and all of the members of the group. I thus arrive at 
a group having the two properties: [a) that all the members of 
the group are compresent, (6) that nothing outside the group is 
compresent with every member of the group. Such a group I 
shall call a “complete complex of compresence”. 

Such a complex I suppose to consist of constituents most of 
which, in the natural course of events, may be expected to be 
members of many other complexes. The shade of colour C, we 
supposed, recurs every time anybody sees a rainbow distinctly. 
My recollection may be qualitatively indistinguishable from a 
recollection that I had yesterday. My apprehension of dental pain 
may be just what I felt before my last visit to the dentist. All these 
items of the complex of compresence may occur frequently, and 
are not essentially dated. That is to say, if A is one of them, and 
A precedes (or follows) B, we have no reason to suppose that 
A and B are not identical. 

Have we any reason, either logical or empirical, to believe that 
a complete complex of compresence, aB a whole, cannot be 
repeated? Let us, in the first place, confine ourselves to the 



person’s experience. My visual field is very complex, though 
probably not infinitely complex. Every time I move my eyes, the 
visual qualities connected with a given object which remains 
visible undergo changes : what I see out of the corner of my eyes 
looks different from what is in the centre of my field of vision. 
If it is true, as some maintain, that my memory is coloured by 
my whole past experience, then it follows logically that my 
total recollections cannot be exactly similar on two different 
occasions; even if we reject this doctrine, such exact similarity 
seems very improbable. 

From such considerations I think we ought to conclude that 
the exact repetition of my total momentary experience, which 
is what, in this connection, I call a “complete complex of corn- 
presence”, is not logically impossible, but is empirically so 
exceedingly improbable that we may assume its non-occurrence. 
In that case, a complete complex of compresence will, so far as 
one person’s experience is concerned, have the formal properties 
required of “events”, i.e.: if A, B, C are complete complexes of 
compresence, then if A wholly precedes B, A and B are not 
identical; and if B also wholly precedes C, then A wholly precedes 
C. We thus have the requisites for defining the time-order in one 
person’s experience. 

This, however, is only part, and not the most difficult part, of 
what we have to accomplish. We have to extend space-time order 
beyond one person’s experience to the experiences of different 
people and to the physical world. In regard to the physical world, 
especially, this is difficult. 

So long as we confine ourselves to one person’s experience, we 
need only concern ourselves with time. But now we have also 
to take account of space. That is to say, we have to find a definition 
of “events” which shall insure that each event has, not merely 
a unique temporal position, but a unique spatio-temporal position. 

So long as we confine ourselves to experiences, there is no 
fresh difficulty of a serious kind. It may be taken as virtually 
certain, on empirical grounds, that my visual field, whenever my 
eyes are open, is not exactly similar to that of any one else. If 
A and B are looking simultaneously at the same scene, there are 
differences of perspective ; if they change places, A will not see 
exactly what B was seeing, because of differences of eyesight, 
changes of lighting meanwhile, and so on. In short, the reasons 


HUMAN knowledge: its scope and limits 

for supposing that no total momentary experience of A is ever 
exactly like some total momentary experience of B are of the 
same kind as the reasons for supposing that no two total momen- 
tary experiences of A are ever exactly alike. 

This being granted, we can establish a spatial order among 
percipients by means of the laws of perspective, provided there 
is any physical object that all the percipients concerned are per- 
ceiving. If there is not, a process by means of intermediate links 
can reach the same result. There are of course complications and 
difficulties, but they are not such as concern our subject at all 
closely, and we may safely ignore them. 

What can be said about the purely physical world is hypo- 
thetical, since physics gives no information except as to structure. 
But there are reasons for supposing that, at every place in physical 
space-time, there is at every moment a multiplicity of occur- 
rences, just as there is in a mind. “Compresence”, which I take 
to have a merely ostensive definition, appears in psychology as 
“simultaneity in one experience ,, ) but in physics as “overlapping 
in space-time”. If, as I maintain, my thoughts are in my head, 
it is obvious that these are different aspects of one relation. How- 
ever, this identification is inessential to my present argument. 

When I look at the stars on a clear night, each star that I see 
has an effect on me, and has an effect on the eye before it has 
an effect on the mind. It follows that, at the surface of the eye, 
something causally connected with each visible star is happening. 
The same considerations apply to ordinary objects seen in day- 
light. At this moment I can see white pages covered with writing, 
some books, an oval table, innumerable chimneys, green trees, 
clouds, and blue sky. I can see these things because there is a 
chain of physical causation from them to my eyes and thence 
to the brain. It follows that what is going on at the surface of 
my eye is as complex as my visual field, in fact as complex as 
the whole of what I can see. This complexity must be physical, 
not merely physiological or psychological; the optic nerve could 
not make the complex responses that it does make except under 
the influence of equally complex stimuli. We must hold that, 
wherever the light of a certain star penetrates, something con- 
nected with that star is happening. Therefore in a place where 
a telescope photographs many millions of stars, many millions 
of things must be happening, each connected with its own star. 



These things are only "experienced” in places where there is a 
recording nervous system, but that they happen in other places 
also can be shown by cameras and dictaphones. There is therefore 
no difficulty of principle in constructing "complexes of corn- 
presence”, where there are no percipients, on the same principles 
as we employed in dealing with momentary experiences. 

Abandoning speculations about the physical world, about which 
our knowledge is very limited, let us return to the world of 
experience. The view which I am suggesting, as preferable to the 
assumption of such wholly colourless particulars as points of space 
or particles of matter, may be expressed as follows : 

There is a relation, which I call "compresence”, which holds 
between two or more qualities when one person experiences them 
simultaneously — for example, between high C and vermilion when 
you hear one and see the other. We can form groups of qualities 
having the following two properties: (a) all members of the 
group are compresent; ( b ) given anything not a member of the 
group, there is at least one member of the group with which it 
is not compresent. Any one such complete group of compresent 
qualities constitutes a single complex whole, defined when its 
constituents are given, but itself a unit, not a class. That is to 
say, it is something which exists, not merely because its con- 
stituents exist, but because, in virtue of being compresent, they 
constitute a single structure. One such structure, when composed 
of mental constituents, may be called a "total momentary 

Total momentary experiences, as opposed to qualities, have 
time relations possessing the desired characteristics. I can see 
blue yesterday, red to-day, and blue again to-morrow. Therefore, 
so far as qualities are concerned, blue is before red and red is 
before blue, while blue, since it occurs yesterday and to-morrow, 
is before itself. We cannot therefore construct, out of qualities 
alone, such a relation as will generate a series. But out of total 
momentary experiences we can do this, provided no total momen- 
tary experience ever exactly recurs. That this does not happen 
is an empirical proposition, but, so far as our experience goes, 
a well-grounded one. I regard it as a merit in the above theory 
that it gets rid of what would otherwise be synthetic a priori 
knowledge. That, if A precedes B, B does not precede A, and 
that, if A precedes B and B precedes C, then A precedes C, are 



synthetic propositions ; moreover, as we have just seen, they are 
not true if A and B and C are qualities. By making such statements 
(in so far as they are true) empirical generalizations, we overcome 
what would otherwise be a grave difficulty in the theory of 

I come back now to the conception of “instance”. An “instance” 
of a quality, as I wish to use the word, is a complex of compresent 
qualities of which the quality in question is one. In some cases 
this view seems natural. An instance of “man” has other qualities 
besides humanity: he is white or black, French or English, wise 
or foolish, and so on. His passport enumerates enough of his 
characteristics to distinguish him from the rest of the human 
race. Each of these characteristics, presumably, exists in many 
other instances. There are baby giraffes who have the height men- 
tioned in his passport, and parrots who have the same birthday 
as he has. It is only the assemblage of qualities that makes the 
instance unique. Every man, in fact, is defined by such an 
assemblage of qualities, of which humanity is only one. 

But when we come to points of space, instants of time, particles 
of matter, and such stock-in-trade of abstract science, we feel as 
if a particular could be a “mere” instance, differentiated from 
other instances by relations, not by qualities. To some degree, 
we think this of less abstract objects: we say “as like as two 
peas”, suggesting that between two peas there are no qualitative 
differences. We think also that two patches of colour may be 
merely two, and may differ only numerically. This way of thinking, 
I maintain, is a mistake. I should say that, when the same shade 
of colour exists in two places at once, it is one, not two; there 
are, however, two complexes, in which the shade of colour is 
combined with the qualities that give position in the visual field. 
People have become so obsessed with the relativity of spatial 
position in physics that they have become oblivious of the 
absoluteness of spatial position in the visual field. At every 
moment, what is in the centre of my field of vision has a quality 
that may be called “centrality”; what is to the right is “dexter”, 
what to the left “sinister”, what above “superior”, what below 
“inferior”. These are qualities of the visual datum, not relations. 
It is the complex consisting of one such quality combined with 
a shade of colour that is distinct from the complex consisting of 
the same shade elsewhere. In short, the multiplicity of instances 

3 j 6 


of a given shade of colour is formed exactly as the multiplicity 
of instances of humanity is formed, namely by the addition of 
other qualities. 

As for points, instants, and particles, in so far as they are not 
logical fictions similar considerations apply. Take first instants. 
It will be found that what I call a “total momentary experience” 
has all the formal properties required of an “instant” in my 
biography. And it will be found that, where there is only matter, 
the “complete complex of compresence” may serve to define an 
instant of Einsteinian local time, or to define a “point-instant” 
in cosmic space-time. Points in perceptual space are defined 
without any trouble, since the qualities of up-and-down, right- 
and-left, in their various degrees, have already all the properties 
that we require of “points”. It is indeed this fact, together with 
perception of depth, that has led us to place such emphasis on 
the spatial characteristics of the world. 

I do not think “particles” can be dealt with quite in the above 
manner. In any case, they are no longer part of the fundamental 
apparatus of physics. They are, I should say, strings of events 
interconnected by the law of inertia. They are no longer indes- 
tructible, and have become merely convenient approximations. 

I come now to a possible objection to the above theory, which 
was advanced by Arnauld against Leibniz. If a “particular” is 
really a complex of qualities, then the statement that such-and- 
such a particular has such-and-such a quality must, when true, 
be analytic; at least, so it would seem. Leibniz held (i) that every 
proposition has a subject and a predicate; (2) that a substance 
is defined by the total of its predicates; (3) that the soul is a 
substance. It followed that everything that can be truly said of 
a given soul consists in mentioning some predicate which is one 
of those that constitute the given soul. “Caesar”, for example, was 
a collection of predicates, one of which was “crossing the Rubi- 
con”. He was therefore compelled by logic to cross the Rubicon, 
and there is no such thing as contingency or free will. Leibniz 
ought, on this point, to have agreed with Spinoza, but he chose 
not to, for reasons discreditable either to his intellect or to his 
moral character. The question is: Can I avoid agreeing with 
Spinoza without equal discredit ? 

What we have to consider is a subject-predicate proposition 
expressing a judgment of perception, such as “this is red”. What 


human knowledge: its scope and limits 

is “this”? Clearly it is not my whole momentary experience; I 
am not saying “one of the qualities that I am at present experienc- 
ing is “redness”. The word “this” may be accompanied by a 
gesture, indicating that I mean what is in a certain direction, say 
the centre of my visual field. In that case, the core of what I am 
saying may be expressed by “centrality and redness overlap 
spatially in my present visual field”. It is to be observed that, 
within the large complex of my total momentary experience, there 
are smaller complexes constituted by spatial compresence in 
perceptual space. Whatever quality I see in a certain direction has 
perceptual-spatial compresence with the visual quality con- 
stituting that direction. It would seem that the word “this”, 
accompanied by a gesture, is equivalent to a description, e.g. 
“what is occupying the centre of my visual field”. To say that 
this description applies to redness is to say something which 
clearly is not analytic. But since it employs a description instead 
of a name, it is not quite what we set out to consider. 

We were considering what sort of thing could have the formal 
properties that are required for space-time order. Such a thing 
must happen in only one time and place ; it must not recur, either 
on another occasion or in another location. So far as time and 
physical space are concerned, these conditions are satisfied by 
the “complete complex of compresence”, whether this consists 
of my momentary experiences or of a full group of overlapping 
physical qualities. (I call such a group “full” when, if anything 
is added, the members will no longer be all compresent.) But 
when we come to consider perceptual space, we have no need 
of an analogous procedure. If I see simultaneously two patches 
of a given shade of colour, they differ as regards the qualities of 
up-and-down, right-and-left, and it is by means of these qualities 
that the patches acquire particularity. 

With these preliminaries, let us examine the question of proper 

It seems preposterous to maintain that “Caesar crossed the 
Rubicon” is an analytic proposition. But if it is not, what do we 
mean by “Caesar”? 

Taking Caesar as he was, without the limitations due to our 
ignorance, we may say that he was a series of events, each event 
being a momentary total experience. If we were to define “Caesar” 
by enumerating these events, the crossing of the Rubicon would 


have to come in our list, and ‘ 4 Caesar crossed the Rubicon” would 
be analytic. But in fact we do not define “Caesar” in this way, and 
we cannot do so, since we do not know all his experiences. What 
happens in fact is more like this: Certain series of experiences 
have certain characteristics which make us call such a series a 
“person”. Every person has a number of characteristics that are 
peculiar to him; Caesar, for example, had the name “Julius Caesar”. 
Suppose P is some property which has belonged to only one 
person; then we can say: “I give the name ‘A* to the person who 
had the property P”. In this case, the name “A” is an abbreviation 
for “the person who had the property P”. It is obvious that, if 
this person also had the property Q, the statement “A had the 
property Q” is not analytic unless Q is analytically a consequence 
of P. 

This is all very well as regards a historical character, but how 
about somebody whom I know more intimately, e.g. myself? How 
about such a statement as “I am hot”? This may, following our 
earlier analysis, be translated into “heat is one of the qualities 
that make up I-now”. Here “I-now” may be taken as denoting 
the same complex that is denoted by “my total present momen- 
tary experience”. But the question remains: how do I know what 
is denoted by “I-now”? What is denoted is continually changing; 
on no two occasions can the denotation be the same. But clearly 
the words “I-now” have in some sense a constant meaning; they 
are fixed elements in the language. We cannot say that, in the 
ordinary sense, “I-now” is a name, like “Julius Caesar”, because 
to know what it denotes we must know when and by whom it is 
used. Nor has it any definable conceptual content, for that, 
equally, would not vary with each occasion when the phrase is 
used. Exactly the same problems arise in regard to the word 

But although “I-now” and “this” are not names in quite the 
ordinary sense, I incline to think that there is a sense in which 
they must count as names. A proper name, as opposed to a 
concealed description, can be given to the whole or to any part 
of what the speaker is at the moment experiencing. When our 
verbal inventiveness fails, we fall back on “this” for the part of 
our total momentary experience to which we are specially,attending, 
and upon “I-now” for the total momentary experience. I maintain 
that I can perceive a complex of compresent qualities without 



necessarily perceiving all the constituent qualities. I can give the 
name “this” to such a complex, and then, by attention, observe 
that redness (say) is one of its constituent qualities. The resulting 
knowledge I express in the sentence “this is red”, which, accord- 
ingly, is a judgment of analysis, but not, in the logical sense, an 
analytic judgment. A complex can be perceived without my being 
aware of all its parts; when, by attention, I become aware that 
it has such-and-such a part, this is a judgment of perception 
which analyses the whole, but is not analytic, because the whole 
was defined as “this”, not as a complex of known parts. 

The kind of thing I have in mind is the kind of thing that is 
emphasized by the Gestalt psychologists. Suppose I possessed 
a clock which showed not only hours and minutes, but the day 
of the month, the month of the year, and the year of the Christian 
era, and suppose that this clock were to function throughout my 
life. It would then never twice during my life present the same 
appearance. I might perceive that two appearances of it were 
different, without being able to say at once in what the difference 
consisted. Attention might lead me to say: “In this appearance 
the minute-hand is at the top ; in that, it is at the bottom”. Here 
“this” and “that” are merely names, and therefore nothing said 
about them can be logically analytic. 

There is another way of escaping from the conclusion that 
judgments are analytic when in fact they are obviously empirical. 
Consider again our clock that never repeats itself. We can define 
a date unambiguously by means of this clock. Suppose that, when 
the clock indicates io hours 47 minutes on June 15, 1947, I say 
“I am hot”. This can be translated into: “Hotness is compresent 
with the appearance of the clock that is described as 10 hours 
47 minutes, June 15, 1947”. This is certainly not analytic. 

One way of making clear the scope and purport of our discussion 
is to put it in terms of “minimum vocabularies”. We may ask: 
“What is, in principle, a minimum vocabulary for describing the 
world of my sensible experience?” We have to ask ourselves: 
Can I be content with names of qualities, and words for corn- 
presence and for spatial and temporal relations, or do I need also 
proper names? And in the latter case, what sorts of things will 
need proper names ? 

I have suggested that ordinary proper names, such as “Socrates”, 
“France”, or “the sun”, apply to continuous portions of space- 



time which happen to interest us, and that space-time is composed 
of “complete complexes of compresence”, which themselves are 
composed of qualities. According to this theory, an “instance” 
of (say) a shade of colour is a complex of which that shade is 
a constituent. The colour itself exists wherever (as we should 
commonly say) there is something that has that colour. Any 
collection of compresent qualities may be called a “complex of 
compresence”, but it is only a “complete complex” when it cannot 
be enlarged without ceasing to be a complex of compresence. Often 
a complete complex can be rendered definite by mentioning only 
some of its components ; e.g. in the above case of the clock, the 
complex is determined when we are told what appearance of the 
clock belongs to it. This is what makes dating convenient. 

Subject-predicate propositions expressing judgments of per- 
ception occur in two ways. First: if a complex is rendered deter- 
minate when only some of its constituent qualities are assigned, 
we may state that this complex also has such-and-such other 
qualities; this is illustrated by the statement “I was hot when the 
clock said 10 hours 47 minutes”. 

Second : I may perceive a complex without being aware of all 
its parts; in that case, I may, by attention, arrive at a judgment 
of perception of the form “P is part of W”, where “W” is the 
proper name of the perceived complex. If such judgments are 
admitted as irreducible, we need proper names for complexes. 
But it would seem that the need for such judgments only arises 
through ignorance, and that, with better knowledge, our whole 
W can always be described by means of its constituents. I think, 
therefore, though with some hesitation, that there is no theoretical 
need for proper names as opposed to names of qualities and of 
relations. Whatever is dated and located is complex, and the 
notion of simple “particulars” is a mistake. 

As the subject of this Chapter is somewhat difficult, it will 
perhaps contribute to clarity and to the prevention of misunder- 
standing to repeat the main points of the above discussion more 
briefly and less controversially. Let us begin with “compresence”. 

“Compresence”, as I wish to understand the term, applies to 
the physical world as well as to the world of mind. In the physical 
world it is equivalent to “overlapping in space-time”, but this 
cannot be taken as its definition, since compresence is needed 
in defining spatio-temporal position. I wish to emphasize that the 



HUMAN knowledge: its scope and limits 

relation is to be the very same in physics as in psychology. Just 
as many things happen simultaneously in my mind, so, we must 
suppose, many things happen simultaneously in every place in 
space-time. When we look at the night sky, each star that we can 
see produces its separate effect, and this is only possible if, at the 
surface of the eye, things are happening that are connected with 
each visible star. These different things are all “compresent”. 

Wherever several things are compresent, they form what I shall 
call a “complex of compresence”. If there are other things com- 
present with all of them, they can be added to form a larger 
complex. When it is no longer possible to find anything com- 
present with all the constituents of the complex, I call the complex 
“complete”. Thus a “complete complex of compresence” is one 
whose constituents have the two properties {a) that all of them 
are compresent, (b) that nothing outside the group is compresent 
with every member of the group. 

“I-now” denotes the complete complex of compresence which 
contains the present contents of my mind. “This” denotes what- 
ever part of this complex I am specially noticing. 

Complete complexes of compresence are the subjects of spatio- 
temporal relations in physical space-time. For empirical, not 
logical, reasons, it is highly probable that none of them recurs, 
i.e. that none of them precedes itself, or is north of itself, or west 
of itself, or above itself. 

A complete complex of compresence counts as a space-time 

A complex which is not complete will, in general, be a part 
of various complete complexes; so will a single quality. A given 
shade of colour, for example, is part of every complete complex 
which is a space-time point at which this shade exists. To say 
of a quality or of an incomplete complex that it “exists at” such- 
and-such a space-time point is to say that it is part of the complete 
complex which is that point. 

An incomplete complex occupies a continuous region in space- 
time if, given any two space-time points of which it is part, there 
is a continuous route, from the one to the other, consisting wholly 
of points of which the incomplete complex is part. 

Such a complex may be called an “event”. It has the property 
of non-recurrence, but not that of occupying only one space-time 



The occupation of a continuous region by a given incomplete 
complex may be defined as follows. A complete complex B is said 
to be “between” two not too distant complete complexes A and C 
if what is common to A and C is part of B. A collection of complete 
complexes is “continuous” (for our purposes) if between any two 
of its members there are other members of the collection. This, 
however, is only a rough-and-ready definition; a precise definition 
could only be given by means of topology. 

We can never know that a given complex of compresence is 
complete, since there may always be something else, of which 
we are not aware, which is compresent with every part of the 
given complex. This is another way of saying that we cannot, 
in practice, define a place or a date exactly. 

Certain incomplete complexes have advantages from the point 
of view of dating. Take, for example, the date in to-day’s news- 
paper together with a 24-hour clock which is going. These two 
together make a complex which never recurs, and of which the 
duration is so brief that for most purposes we do not need to 
notice that it is more than an instant. It is by means of such 
incomplete complexes that we in fact determine dates. 

For determining spatial position, there are similar advantages 
in the ocular qualities of centrality, up-and-down, and right-and- 
left. These qualities are mutually exclusive as regards what may 
be called “private compresence”, which is a relation between 
elements in one total momentary experience. The quality of 
centrality, for example, has “private compresence” with the colour 
which is occupying the centre of my visual field. The correlation 
of places in my private space with places in physical space pro- 
ceeds on the assumption that, if visual percepts are not privately 
compresent, the corresponding physical objects are not publicly 
compresent, but if the visual percepts are privately compresent, 
the corresponding physical objects may differ in distance from 
the percipient, though they will agree approximately in direction. 
Thus private compresence of percepts is a necessary but not 
sufficient condition for public compresence of the corresponding 
physical objects. 

It is to be observed that, in general, every increase in the number 
of qualities combined in a complex of compresence diminishes 
the amount of space- time that it occupies. A complete complex 
of compresence will occupy a portion of space-time which has 


human knowledge: its scope and limits 

no parts that are portions of space-time ; if we assume continuity, 
such a portion will have the properties that we expect of a point- 
instant. But there is no reason, either empirical or a priori , to 
suppose either that space-time is continuous, or that it is not; 
everything known can be explained equally well on either hypo- 
thesis. If it is not continuous, a finite number of complexes of 
compresence will occupy a finite space-time volume, and the 
structure of space-time will be granular, like that of a heap of 

A complex of compresence, as I conceive it, is determinate 
when the qualities constituting it are given. That is to say, if the 
qualities q l9 q 2> . . . q Q are all mutually compresent, there is just 
one complex of compresence, say C, which consists of the com- 
bination of these qualities. It is always logically possible for C 
to occur more than once, but I assume that, if C is sufficiently 
complex, there will not in fact be recurrence. A few words are 
necessary to explain what, logically, is meant by “recurrence”. 
Let us, for simplicity, confine ourselves to time in one biography, 
and let us begin by considering complete complexes. 

I assume that, between any two complete complexes belonging 
to the same biography, there is a relation of earlier-and-later. 
To suppose that a complete complex can recur is to suppose that 
a complete complex can have the relation of earlier-and-later to 
itself. This, I assume, does not happen, or at any rate does not 
happen within any ordinary period of time. I do not mean to 
deny dogmatically that history may be cyclic, as some Stoics 
thought, but the possibility is too remote to need to be taken 
into account. 

Since we can never know that a known complex of compresence 
is complete — since, in fact, we can be pretty sure that it is not — 
we use, in practice, for purposes of chronology and geography, 
such incomplete complexes as either do not recur at all, or recur 
in a fairly regular manner. The date on a calendar persists for 
twenty-four hours, and then changes abruptly. Some clocks have 
a minute-hand that moves with a jerk once a minute; the appear- 
ance of such a clock persists for a minute and recurs every twelve 
hours. If we had sixty such clocks in a circle, and each gave its 
jerk one second after the one to the left of it, the complex con- 
sisting of the appearance of all the sixty would fix the time within 
one second. By such methods accuracy of dating may be indefi- 



nitely increased. Exactly similar remarks apply to methods of 
determining latitude and longitude. 

A complex of compresence, though defined when all its con- 
stituent qualities are given, is not to be conceived, like a class, 
as a mere logical construction, but as something which can be 
known and named without our having to know all its constituent 
qualities. The logical point involved may be made clear as follows: 
the relation earlier-and-later holds, primarily, between two com- 
plete complexes of compresence, and only in a derivative and 
definable sense between partial complexes. In the case of a purely 
logical structure, a statement about the structure can be reduced 
to one about its components, but in the case of the time-order 
this is not possible on the theory of “particulars” adopted in this 
Chapter. A complex can, therefore, be mentioned in a way which 
is not reducible to a statement about any or all of its constituents. 
It is, in fact, the sort of object that is a “this”, and that can have 
a proper name. A given collection of qualities only forms a com- 
plex of compresence if the qualities happen to be all mutually 
compresent; when they are, the complex is something new, over 
and above the qualities, though necessarily unique when the 
qualities are given. 

According to the above theory, a complex of compresence which 
does not recur takes the place traditionally occupied by “particu- 
lars”; a single such complex, or a string of such complexes 
causally connected in a certain way, is the kind of object to which 
it is conventionally appropriate to give a proper name. But a 
complex of compresence is of the same logical type as a single 
quality, that is to say, any statement which is significant about 
either is significant, though probably not true, about the other. 

We may agree with Leibniz to this extent, that only our 
ignorance makes names for complexes necessary. In theory, every 
complex of compresence can be defined by enumerating its com- 
ponent qualities. But in fact we can perceive a complex without 
perceiving all its component qualities ; in this case, if we discover 
that a certain quality is a component of it, we need a name for 
the complex to express what it is that we have discovered. The 
need for proper names, therefore, is bound up with our way of 
acquiring knowledge, and would cease if our knowledge were 

3 2 S 

Chapter IX 

T HE practical utility of science depends upon its ability to 
foretell the future. When the atomic bombs were dropped, 
it was expected that large numbers of Japanese would die, 
and they did. Such highly satisfactory results have led, in our day, 
to an admiration of science, which is due to the pleasure we derive 
from the satisfaction of our lust for power. The most powerful 
communities are the most scientific, though it is not the men of 
science who wield the power conferred by their knowledge. On 
the contrary, the actual men of science are rapidly sinking into the 
position of state prisoners, condemned to slave labour by brutal 
masters, like subject djinns in the Arabian Nights. But we must 
not waste any more time upon such pleasant topics. The power of 
science is due to its discovery of causal laws, and it is causal laws 
that are to occupy us in this chapter. 

|^A “causal law”, as I shall use the term, may be defined as a 
general principle in virtue of which, given sufficient data about 
certain regions of space-time, it is possible to infer something 
about certain other regions of space-time. The inference may be 
only probable, but the probability must be considerably more than 
a half if the principle in question is to be considered worthy to be 
called a “causal law”. 

I have purposely made the above definition very wide. In the 
first place, the region to which we infer need not be later than those 
from which we infer. There are, it is true, some laws — notably 
the second law of thermodynamics — which allow inferences for- 
wards more readily than backwards, but this is not a general 
characteristic of causal laws. In geology, for example, the inferences 
are almost all backwards. In the second place, we cannot lay 
down rules as to the number of data that may be involved in 
stating a law. If it should ever become possible to state the laws 
of embryology in terms of physics, enormously complex data 
would be required. In the third place, the inference may be only 
to some more or less general characteristic of the inferred event 
or events. In the days before Galileo it was known that un- 
supported heavy bodies fall, which was a causal law; but it was 



not known how fast they fall, so that when a weight was dropped 
it was impossible to say accurately where it would be after a given 
lapse of time. In the fourth place, if the law states a high degree 
of probability it may be almost as satisfactory as if it stated a 
certainty. I am not thinking of the probability of the law being 
true ; causal laws, like the rest of our knowledge, may be mistaken. 
What I am thinking of is that some laws state probabilities, for 
example the statistical laws of quantum theory. Such laws, 
supposing them completely true, make inferred events only 
probable, but this does not prevent them from counting as causal 
laws according to the above definition. 

One advantage of admitting laws which only confer probability 
is that it enables us to incorporate in science the crude generaliza- 
tions from which common sense starts, such as “fire burns”, 
“bread nourishes”, “dogs bark”, or “lions are fierce”. All these 
are causal laws, and all are liable to exceptions, so that in a given 
case they confer only probability. The fire on a plum pudding 
does not burn you, poisoned bread does not nourish, some dogs 
are too lazy to bark, and some lions grow so fond of their keepers 
that they cease to be fierce. But in the great majority of cases the 
above generalizations will be a sound guide in action. There are 
a large number of such approximate regularities which are assumed 
in our every-day behaviour, and it is from them that the conception 
of causal laws arose. Scientific laws, it is true, are no longer so 
simple: they have become complicated in the endeavour to give 
them a form in which they are not liable to exceptions. But the 
old simpler laws remain valid so long as they are only regarded 
as asserting probabilities. 

Causal laws are of two sorts, those concerned with persistence 
and those concerned with change. The former kind are often not 
regarded as causal, but this is a mistake. A good example of a law 
of persistence is the first law of motion. Another example is the 
persistence of matter. After the discovery of oxygen, when the 
process of combustion came to be understood, it was possible to 
regard all matter as indestructible. It has now become doubtful 
whether this is quite true, but it remains true for most practical 
purposes. What appears to be more exactly true is the persistence 
of energy. The gradual development of laws stating persistence 
started from the common- sense belief, based on pre-scientific 
experience, that most solid objects continue to exist until they 


human knowledge: its scope and limits 

crumble from old age or are destroyed by fire, and that, when this 
happens, it is possible to suppose that their small parts survive 
in a new arrangement. It was this pre-scientific point of view that 
gave rise to the belief in material substance. 

Causal laws concerned with change were found by Galileo and 
Newton to demand statement in terms of acceleration, i.e. change 
of velocity in magnitude or direction or both. The greatest 
triumph of this point of view was the law of gravitation, according 
to which every particle of matter causes in every other an accelera- 
tion directly proportional to the mass of the attracting particle 
and inversely proportional to the square of the distance between 
them. But Einstein's form of the law of gravitation made it more 
analogous to the law of inertia, and, in a sense, a law of persistence 
rather than a law of change. According to Einstein, space-time is 
full of what we may call hills ; each hill grows steeper as you go 
up, and has a piece of matter at the top. The result is that the 
easiest route from place to place is one which winds round the hills. 
The law of gravitation consists in the fact that bodies always take 
the easiest route, which is what is called a “geodesic". There is a 
law of cosmic laziness called the “principle of least action", which 
states that when a body moves from one place to another it will 
choose the route involving least work. By means of this principle 
gravitation is absorbed into the geometry of space-time. 

The essential laws of change in modern physics are those of 
quantum theory, which govern transitions from one form of 
energy to another. An atom can emit energy in the form of light, 
which then travels on unchanged until it meets another atom, 
which may absorb the energy of the light. Such interchanges are 
governed by certain rules, which do not suffice to say what will 
happen on a given occasion, but can predict, with a very high 
degree of probability, the statistical distribution of possible 
happenings among a very large number of interchanges. This is 
as near as physics can get at present to the ultimate character of 
causal laws. 

Everything that we believe ourselves to know about the physical 
world depends entirely upon the assumption that there are 
causal laws. Sensations, and what we optimistically call “percep- 
tions", are events in us. We do not actually see physical objects, 
any more than we hear electromagnetic waves when we listen to 
the wireless. What we directly experience might be all that exists, 



if we did not have reason to believe that our sensations have 
external causes. It is important, therefore, to inquire into our 
belief in causation. Is it mere superstitition, or has it a solid 
foundation ? 

The question of the justification of our belief in causality be- 
longs to theory of knowledge, and I shall therefore postpone it 
for the present. My purpose in this Part is the interpretation of 
science, not an inquiry into the grounds for supposing science 
valid. Science assumes causality in some sense, and our present 
question is: in what sense is causality involved in scientific 
method P 1 

Broadly speaking, scientific method consists in inventing 
hypotheses which fit the data, which are as simple as is compatible 
with this requirement, and which make it possible to draw infer- 
ences subsequently confirmed by observation. The theory of 
probability shows that the validity of this process depends upon 
an assumption which may be roughly stated as the postulate that 
there are general laws of certain kinds. This postulate, in a 
suitable form, can make scientific laws probable, but without it 
they do not even achieve probability. We have therefore to 
examine this assumption, to find out the most plausible form in 
which it is both effective and possibly true. 

If there is no limit to the complexity of possible laws, every 
imaginable course of events will be subject to laws, and therefore 
the assumption that there are laws will become a tautology. Take, 
for example, the numbers of all the taxis that I have hired in the 
course of my life, and the times when I have hired them. We have 
here a finite set of integers and a finite number of corresponding 
times. If n is the number of the taxi that I hired at the time t y it 
is certainly possible, in an infinite number of ways, to find a 
function / such that the formula 

n =f(t) 

is true for all the values of n and t that have hitherto occurred. 
An infinite number of these formulae will fail for the next taxi 
that I hire, but there will still be an infinite number that remain 
true. By the time I die, it will be possible to close the account, and 
there will still remain an infinite number of possible formulae, 

1 The following pages anticipate, in an abbreviated form, the fuller 
discussions of Parts V and VI. 


HUMAN knowledge: its scope and limits 

each of which might claim to be a law connecting the number of 
a taxi with the time when I hire it. 

The merit of this example, for my present purpose, is its 
obvious absurdity. In the sense in which we believe in natural 
laws we should say that there is no law connecting the n and / of 
the above formula, and that, if any suggested formula happens to 
work, that is a mere chance. If we had found a formula that 
worked in all cases up to the present, we should not expect it to 
work in the next case. Only a superstitious person whose emotions 
are involved will believe an induction of this sort; gamblers at 
Monte Carlo practise inductions which no man of science would 
sanction. But it is not altogether easy to state the difference be- 
tween the inductions of the superstitious gambler and the in- 
ductions of the prudent man of science. Obviously there is a 
difference, but in what does it consist ? And is the difference such 
as to affect logical validity, or does it consist merely in a difference 
as to the obviousness of the appeal to the emotions ? Is the faith in 
scientific method merely the scientist’s superstition appropriate 
to his kind of gambling? These questions, however, belong to the 
theory of knowledge. For the present I want to discover not why 
we believe, but what we believe, when we believe in natural 

It is customary to speak of induction as what is needed to make 
the truth of scientific laws probable. I do not think that induction, 
pure and simple, is fundamental. The above example of the 
numbers of taxis illustrates this. All past observations as to these 
numbers are compatible with a number of laws of the form 
n — f (t), and these will, as a rule, give different values for the 
next n. We cannot therefore use them all for prediction, and in 
fact we have no inclination to believe in any of them. Generalizing, 
we may say: Every finite set of observations is compatible with a 
number of mutually inconsistent laws, all of which have exactly 
the same inductive evidence in their favour. Therefore pure 
induction is invalid, and is, moreover, not what we in fact believe. 

Whenever inductive evidence seems to us to make a suggested 
law very probable, the law is one which had suggested itself more 
or less independently of the evidence, and had seemed to us in 
some way likely to be true. When this is the case, subsequent 
confirmatory evidence is found astonishingly convincing. 

This, however, is only partially true. If a law is suggested of 



which the consequences are very different from what we should 
expect, and it then is confirmed by observation, we are more prone 
to believe in it than if its results were commonplace. But in such 
a case the law itself may seem plausible, although its consequences, 
when mentioned, are found surprising. Perhaps one of the most 
important effects of scientific education is to modify the hypo- 
theses that appear prima facie probable. It was this cause, not direct 
negative evidence, that led the belief in witchcraft to decay. If you 
had a number of outwardly similar boxes, of which some contained 
gyrostats, and you showed them to a savage, saying that by 
uttering a magic formula you could make any one of them im- 
possible to turn round, the inductive evidence would soon per- 
suade him that you were right, but a man of educated scientific 
outlook would search for some other explanation in spite of 
repeated apparent verifications of your “law”. 

Induction, moreover, does not validate many of the inferences 
in which science feels most confidence. We are all convinced that, 
when a number of people hear a sound simultaneously, their 
common experience has an external source, which is propagated 
through the intervening medium by sound waves. There cannot 
be inductive evidence (unless in some extended sense) for some- 
thing outside human experience, such as a sound wave. Our 
experience will be the same whether there really are sound-waves, 
or, though there are none, auditory sensations occur as they 
would if there were sound-waves ; no inductive evidence can ever 
favour one of these hypotheses rather than the other. Nevertheless 
every one in fact accepts the realist alternative — even the idealist 
philosopher except in his professional moments. We do this on 
grounds that have nothing to do with induction — partly because 
we like laws to be as simple as possible, partly because we believe 
that causal laws must have spatio-temporal continuity, i.e. must 
not involve action at a distance. 

In the establishment of scientific laws experience plays a two- 
fold part. There is the obvious confirming or confuting of a hypo- 
thesis by observing whether its calculated consequences take 
place, and there is the previous experience which determines 
what hypotheses we shall think antecedently probable. But behind 
these influences of experience there are certain vague general 
expectations, and unless these confer a finite a priori probability 
on certain kinds of hypotheses, scientific inferences are not valid. 



In clarifying scientific method it is essential to give as much 
precision as possible to these expectations, and to examine whether 
the success of science in any degree confirms their validity. After 
being made precise the expectations are, of course, no longer 
quite what they were while they remained vague, but so long as 
they remain vague the question whether they are true or false is 
also vague. 

It seems to me that what may be called the “faith” of science 
is more or less of the following sort: there are formulae (causal 
laws) connecting events, both perceived and unperceived; these 
formulae exhibit spatio-temporal continuity, i.e. involve no direct 
unmediated relation between events at a finite distance from each 
other; a suggested formula having the above characteristics be- 
comes highly probable if, in addition to fitting in with all past 
observations, it enables us to predict others which are subse- 
quently confirmed and which would be very improbable if the 
formula were false. 

The justification of this “faith”, if any, belongs to theory of 
knowledge. Our present task is completed in having stated it. But 
there is still need of some discussion as to the origin and growth 
of this “faith”. 

There are various possible postulates which can be taken as 
the basis of scientific method, but it is difficult to state them with 
the necessary precision. There is the law of causality; there is the 
uniformity of nature ; there is the reign of law : there is the belief 
in natural kinds, and Keynes’s principle of limited variety; and 
there is structural constancy with spatio-temporal continuity. It 
ought to be possible, out of all these somewhat vague assumptions, 
to distil some definite axiom or axioms which, if true, will confer 
the desired degree of probability on scientific inferences. 

The principle of causality appears in the works of almost all 
philosophers in an elementary form which it never takes in any 
advanced science. They suppose science to assume that, given any 
suitable class of events A, there is always some other class of 
events B such that every A is “caused” by a B; moreover every 
event belongs to some such class. 

Most philosophers have held that “cause” means something 
different from “invariable antecedent”. The difference may be 
illustrated by Geulincx’s two clocks, which both keep perfect 
time ; when one points to the hour, the other strikes, but we do not 



think that the one has “caused” the other to strike. A non- 
scientific Fellow of my College lately remarked in despair: “The 
barometer has ceased to have any effect on the weather”. This was 
felt to be a joke, but if “cause” meant “invariable antecedent” it 
would not be. It is supposed that when A is caused by B the 
sequence is not merely a fact, but is in some sense necessary. This 
conception is bound up with the controversy about free will and 
determinism, summed up by the poet in the following lines : 

There was a young man who said : Damn ! 

I learn with regret that I am 
A creature that moves 
In predestinate grooves, 

In short, not a bus, but a tram. 

As against this view most empiricists have held that “cause” 
means nothing but “invariable antecedent”. The difficulty of this 
view, and indeed of any suggestion that scientific laws are of the 
form “A causes B”, is that such sequences are seldom invariable, 
and, even if they are invariable in fact, circumstances can easily 
be imagined which would prevent them from being so. As a rule, 
if you tell a man he is a silly fool he will be angry, but he may be a 
saint, or may happen to die of apoplexy before he has time to lose 
his temper. If you strike a match on a box it usually lights, but 
sometimes it breaks or is damp. If you throw a stone in the air it 
usually falls down again, but it may be swallowed by an eagle 
under the impression that it is a bird. If you will to move your 
arm it usually moves, but not if you are paralysed. In such ways 
all laws of the form “A causes B” are liable to exceptions, since 
something may intervene to prevent the expected result. 

Nevertheless, there are reasons, of which the strength will 
appear in Part VI, for admitting laws of the form “A causes B”, 
provided that we do so with suitable safeguards and limitations. 
The concept of more or less permanent physical objects, in its 
common-sense form, involves “substance”, and when “substance” 
is rejected we have to find some other way of defining the identity 
of a physical object at different times. I think this must be done 
by means of the concept “causal line”. I call a series of events a 
“causal line”, if given some of them, we can infer something about 
the others without having to know anything about the environment. 
For example, if my doors and windows are shut, and at intervals 


HUMAN knowledge: its scope and limits 

I notice my dog asleep on the hearthrug, I infer that he was there, 
or at least somewhere in the room, at the times when I was not 
noticing him. A photon which travels from a star to my eye is a 
series of events obeying an intrinsic law, but ceasing to obey this 
law when it reaches my eye. When two events belong to one 
causal line, the earlier may be said to “cause” the later. In this 
way laws of the form 44 A causes B” may preserve a certain 
validity. They are important in connection both with perception 
and with persistence of material objects. 

It is the possibility of something intervening that has led 
physics to state its laws in the form of differential equations, 
which may be regarded as stating what is tending to happen. And 
as already explained, classical physics, when presented with 
several causes acting simultaneously, represents the resultant as 
a vector sum, so that, in a sense, each cause produces its effect as 
if no other cause were acting. But in fact the whole conception 
of “cause” is resolved into that of “law”. And laws, as they occur 
in classical physics, are concerned with tendencies at an instant. 
What actually happens is to be inferred by taking the vector sum 
of all the tendencies at an instant, and then integrating to find 
out the result after a finite time. 

All empirical laws are inferred from a finite number of observa- 
tions, eked out by interpolation and extrapolation. The part 
played by interpolation is not always adequately realized. Take 
for example the apparent motions of the planets. We assume that, 
during the day-time, they pursue a smooth course which fits in 
easily with their observed courses during the preceding and 
succeeding nights. It would be a possible hypothesis that planets 
only exist when they are observed, but this would make the laws 
of astronomy very complicated. If it is objected that planets can 
be photographed fairly continuously, the same problem arises as 
regards the photographs : do they exist when no one is looking at 
them? This again is a question of interpolation, and the interpola- 
tion is justified by the fact that it gives the simplest laws com- 
patible with what has been observed. 

Exactly the same principle applies to extrapolation. Astronomy 
makes assertions, not only about what planets have done at all 
times since there were astronomers, but about what they will do 
and what they did before there was any one to notice them. This 
extrapolation is often spoken of as if it involved some principle 



other than that involved in interpolation, but in fact the principle 
is one and the same : to choose the simplest law that fits the known 

As a postulate, however, this is open to grave objections. 
“Simple” is a vague conception. Moreover it often happens that 
a simple law turns out, after a time, to be too simple, and that the 
correct law is more complicated. But in such cases the simple law 
is usually approximately right. If, therefore, we only assert that 
a law is approximately right, we cannot be convicted of error 
when some other law is found to be a still better approximation. 

The uniformity of nature, which is a principle sometimes 
invoked, has no definite meaning except in connection with 
natural laws. If it is already granted that there are natural laws, 
the principle of the uniformity of nature states that time and place 
must not appear explicitly in the formulation of laws: the laws 
must be the same in one part of space- time as in another. This 
principle may or may not be true, but in any case it is insufficient 
as a postulate, since it presupposes the existence of laws. 

The existence of natural kinds underlies most pre-scientific 
generalizations, such as “dogs bark” or “wood floats”. The 
essence of a “natural kind” is that it is a class of objects all of 
which possess a number of properties that are not known to be 
logically interconnected. Dogs bark and growl and wag their 
tails, while cats mew and purr and lick themselves. We do not 
know why all the members of an animal species should share so 
many common qualities, but we observe that they do, and base 
our expectations on what we observe. We should be amazed if 
a cat began to bark. 

Natural kinds are not only of biological importance. Atoms and 
molecules are natural kinds; so are electrons, positrons, and 
neutrons. Quantum theory has introduced a new form of natural 
kinds in its discrete series of energy levels. It is now possible to 
conceive the ultimate structure of the physical world not as a 
continuous flux, in the manner of conventional hydrodynamics, 
but in a more Pythagorean fashion, in which models are derived 
from analogy with a heap of shot. Evolution, which in Darwin’s 
time “broadened slowly down from precedent to precedent”, 
now takes revolutionary leaps by means of mutants, or freaks. 
Perhaps wars and revolutions have made us impatient of gradual- 
ness; however that may be, modem scientific theories are much 


HUMAN knowledge: its scope and limits 

more jolty and jagged than the smooth cosmic stream of ordered 
progress imagined by the Victorians. 

The bearing of all this on induction is of considerable importance. 
If you are dealing with a property which is likely to be character- 
istic of a natural kind, you can generalize fairly safely after very 
few instances. Do seals bark? After hearing half a dozen do so, 
you confidently answer “yes”, because you are persuaded in 
advance that either all seals bark or no seals bark. When you 
have found that a few pieces of copper are good conductors of 
electricity, you unhesitatingly assume that this is true of all 
copper. In such cases a generalization has a finite a priori proba- 
bility, and induction is less precarious than in other problems. 

Keynes has a postulate by which, in his opinion, inductive 
arguments might be justified ; he calls it the principle of limited 
variety. It is a form of the assumption of natural kinds. This is 
one of the expedients in the way of a general assumption which, 
if true, validates scientific method. I shall have more to say about 
it at a later stage. What has been said in this chapter is only by 
way of anticipation. 


Chapter X 


P hysical events are arranged by physics in a four-dimen- 
sional manifold called space-time. This manifold is an 
improvement on the older manifold of “things” arranged 
in varying spatial patterns at varying times ; and this, in turn, was 
an improvement upon the manifold resulting from assuming an 
accurate correspondence between percepts and “things”. No 
doubt physics would like to forget its early history, which, like 
that of many established institutions, is not so creditable as could 
be wished. But unfortunately its title to our allegiance is difficult 
to disentangle from its early association with naive realism ; even 
in its most sophisticated form, it still appears as an emendation, 
for which naive realism supplies the text. 

Perceptual space is a common- sense construction, composed of 
diverse raw materials. There are visual space- relations : up-and- 
down, right-and-left, depth up to a certain distance (after which 
differences of depth become imperceptible). There are the differ- 
ences in sensations of touch which enable us to distinguish a 
touch on one part of the body from a touch on another. There is 
the somewhat vague power of estimating the direction of a sound. 
Then there are experienced correlations, of which the most 
important is the correlation of sight and touch ; there are observa- 
tions of movement, and the experience of moving parts of our 
own body. 

Out of such raw materials (the above list does not claim to 
be complete) common sense constructs a single space containing 
objects perceived and unperceived, the perceived objects being 
identified with percepts, according to the principles of naive 
realism. The unperceived objects, for common sense, are those 
which we should perceive if we were in the right position and 
with suitably adjusted sense-organs, together with objects only 
perceived by others, and objects, such as the interior of the 
earth, which are perceived by no one but inferred by common 

In the passage from the common-sense world to that of physics, 
certain common-sense assumptions are retained, though in a 



HUMAN knowledge: its scope and limits 

modified form. For instance, we assume that the furniture of our 
room continues to exist when we do not see it. Common sense 
supposes that what continues is just what we see when we look, 
but physics says that what continues is the external cause of what 
we see, i.e. a vast assemblage of atoms undergoing frequent 
quantum transformations. In the course of these transformations 
they radiate energy, which, when it comes in contact with a 
human body, has various effects, some of which are called “per- 
ceptions”. Two simultaneous parts of one visual percept have a 
certain visual spatial relation which is a component of the total 
percept; the physical objects which correspond to these parts of 
my total percept have a relation roughly corresponding to this 
visual spatial relation. When I say that the relation “corresponds”, 
I mean that it is part of a system of relations having, to some 
extent, the same geometry as that of visual percepts, and that the 
location of physical objects in physical space has discoverable 
relations to the location of perceptual objects in perceptual space. 

But this correspondence is by no means exact. Let us take, to 
simplify our problem, the heavenly bodies as they are and as they 
appear. As they appear, they do not obviously differ as regards 
distance from us; they look like bright points or patches on the 
celestial sphere. That is to say, their position in visual space is 
defined by only two co-ordinates. But eclipses and occultations 
soon led to the view that they are not in fact all equi-distant from 
the earth, though it was a long time before differences of distance 
among the fixed stars were admitted. To fix the position of a 
heavenly body relatively to ourselves we need three polar co- 
ordinates, r, 6 , <f>. It was assumed that 9 and <f> could be the same 
for the physical star as for the perceived star, but r must be 
computed; in fact a great deal of astronomy has been concerned 
with computing r. The assumption that 6 and <f> are the same in 
visual and physical space is equivalent to the assumption that 
light travels in straight lines. This assumption, after a time, came 
to be thought not exactly true, but it is still sufficiently true for 
a first approximation. 

The 6 and <f> of astronomical space, though they have approxi- 
mately the same numerical measure as the 6 and <f> of visual 
space, are not identical with the latter. If they were identical, the 
hypothesis that light does not move exactly in straight lines would 
be meaningless. This illustrates at once the connection and the 

33 8 


difference between visual space when we look at the night sky 
and astronomical space as constructed by the astronomers. The 
connection is kept as close as may be, but beyond a point it has to 
be abandoned if we are to believe in comparatively simple laws 
governing the real and apparent movements of the heavenly 

Small distances from ourselves are not estimated by the elaborate 
methods required in astronomy. We can roughly “see” small 
distances, though the stereoscope produces this effect deceptively. 
We judge things that touch our body to be close to the part they 
touch. When things are not touching us, we sometimes can move 
so as to come in contact with them; the amount of movement 
required measures, roughly, their initial distance from us. We 
have thus three common-sense ways of estimating the distance of 
visual objects on the surface of the earth. Scientific ways of 
estimating distance use these ways as their foundation, but 
correct them by means of physical laws inferred by assuming them. 
The whole process is one of tinkering. If common-sense estimates 
of distances and sizes are roughly correct, then certain physical 
laws are roughly correct. If these laws are quite correct, the 
common-sense estimates must be slightly amended. If the various 
laws are not exactly compatible, they must be adjusted until the 
inconsistency ceases. Thus observation and theory interact; 
what, in scientific physics, is called an observation is usually 
something involving a considerable admixture of theory. 

Let us now abandon the consideration of the stages towards 
theoretical physics, and compare the finished physical world with 
the world of common sense. I see, let us suppose, a buttercup and 
a bluebell; common sense says the buttercup is yellow and the 
bluebell is blue. Physics says that electromagnetic waves of many 
different frequencies start from the sun and reach the two flowers ; 
when they reach them, the buttercup scatters the waves whose 
frequency produces a yellow sensation, and the bluebell those 
that produce a blue sensation. This difference in the effect of the 
two flowers is assumed to be due to some difference in their 
structure. Thus although yellow and blue exist only where there 
is an eye, the difference between them allows us to infer differences 
between the physical objects in the directions in which we see 
yellow and blue respectively. 

Common sense constructs a single space containing “things” 



which combine properties revealed by different senses, such as hot 
and hard and bright. These “things” are placed by common 
sense in a three-dimensional space, in which distance cannot be 
estimated by common-sense methods unless it is small. Physics 
until recently retained something like “things”, but called it 
“matter”, and robbed it of all properties except position in space. 
The position of a piece of matter in space was roughly identical 
with that of the corresponding “thing”, except that the distance, 
if great, had to be calculated by rather elaborate scientific methods. 

In this picking and choosing among common-sense beliefs, 
physics has acted without formulated principles, but nevertheless 
on a subconscious plan which we must try to make explicit. Part 
of this plan is to retain always as much of the common-sense 
world as is possible without intolerable complication; another 
part is to make such non-refutable assumptions as will lead to 
simple causal laws. This latter procedure is already implicit in 
the common-sense belief in “things”: we do not believe that the 
visible world ceases to exist when we shut our eyes, and we hold 
that the cat exists when it is secretly stealing the cream as well as 
when we are punishing it for doing so. All this is “probable” 
inference: it is logically possible to suppose that the world consists 
only of my percepts, and the inference to the common-sense 
world, as to that of physics, is non- demonstrative. But I do not 
wish to go behind common sense at present; I wish only to 
consider the transition from common sense to physics. 

Modern physics is further from common sense than the physics 
of the nineteenth century. It has dispensed with matter, substi- 
tuting series of events ; it has abandoned continuity in microscopic 
phenomena; and it has substituted statistical averages for strict 
deterministic causality affecting each individual occurrence. But 
it has still retained a great deal of which the source is common 
sense. And there are still continuity and determinism so far as 
macroscopic phenomena are concerned, and for most purposes 
there is still matter. 

The world of physics contains more than the world of percepts, 
and in some respects contains more than the world of common 
sense. But while it exceeds both in quantity, it falls short of both 
in known qualitative variety. Both common sense and physics 
supplement percepts by the assumption that things do not cease 
to exist when unperceived, and by the further assumption that 



things never perceived can often be inferred. Physics supplements 
the common-sense world by the whole theory of microscopic 
phenomena; what it asserts about atoms and their history sur- 
passes what common sense allows itself to infer. 

There are two specially important kinds of chains of events: 
first, those which constitute the history of a given piece of matter; 
second, those which connect an object with the perception of it. 
The sun, for instance, has a biography consisting of all that happens 
in the part of space-time that it occupies ; this biography may be 
said to be the sun. It also emits radiations, some of which reach 
eyes and brains, and cause the sort of occurrence which is called 
“seeing the sun”. Broadly speaking, the former set of events 
consists of quantum transitions, the latter of radiant energy. 
There are correspondingly two sets of causal laws, one set con- 
necting events belonging to the same piece of matter, the other 
connecting parts of the same radiation. There is also a third set of 
laws, concerning the transition from energy in the atom to radiant 
energy and vice versa. 

Perceiving, as we know it introspectively, appears to be some- 
thing quite different from the events that physics considers. There- 
fore if there is to be inference from percepts to physical occurrences, 
or from physical occurrences to percepts, we need laws which, 
prima facie , are not physical. I incline to think that physics can 
be so interpreted as to include these laws, but for the present I 
shall not consider this possibility. Our problem is, therefore: 
taking percepts as we know them in experience, and physical 
occurrences as asserted by physics, what laws do we know that 
inter-connect the two and therefore allow inference from one to 
the other ? 

In part, the answer is already patent to common sense. We see 
when light strikes the eye, we hear when sound strikes the ear, we 
have sensations of touch when the body is in contact with some- 
thing else, and so on. These laws are not laws of physics or physio- 
logy, unless physics is subjected to a radical re- interpretation. 
They are laws stating the physical antecedents of perceptions. 
These antecedents are partly outside the percipient’s body 
(except when he is perceiving something in his own body), partly 
in his sense-organs and nerves, partly in his brain. A failure in any 
of these antecedents prevents the perception. But conversely, 
if one of the later antecedents is caused in an unusual way, the 


human knowledge: its scope and limits 

percept will be what it would have been if the causation had been 
usual, and the percipient is liable to be deceived — for example by 
something seen in a mirror or heard on the wireless, if he is un- 
accustomed to mirrors and wireless. 

Each single inference from a perception to a physical object is 
therefore liable to be erroneous in the sense of causing expectations 
that are not fulfilled. It will not usually be erroneous in this sense, 
since the habit of making that sort of inference must have been 
generated by a number of occasions when the inference was 
justified. But here a little further precision is necessary. From a 
practical point of view, an inference from a percept is justified 
if it gives rise to expectations that are verified. This, however, is 
all within the realm of percepts. All that strictly follows is that 
our inferences as to physical objects are consistent with exper- 
ience, but there may be other hypotheses that are equally con- 

The justification of our inferences from perception to physical 
objects depends upon the consistency of the whole system. First, 
from ordinary perceptions, we arrive at an elementary kind of 
physics; this suffices to cause us to put in a separate category 
dreams, mirages, etc., which contradict our elementary physics. 
We then set to work to improve our elementary physics so as to 
include the exceptional phenomena; there is, for instance, a 
perfectly good physical theory of mirages. We learn in this way to 
be critical, and we form the concept of a “trained observer”. We 
are critical of percepts in the name of laws, and of laws in the 
name of percepts; gradually, as physics improves, a closer and 
closer harmony between percepts and laws is established. 

But when I say that we become critical of percepts, I must 
guard against a misunderstanding. Percepts certainly occur, and a 
theory which has to deny any of them is faulty ; but some, being 
caused in an unusual way, lead common sense into erroneous 
inferences. Of this the mirage is a good example. If I see a lake 
which is only a mirage, I see what I see just as truly as if there 
were a physical lake; I am mistaken, not as to the percept, but 
as to what it implies. The percept makes me think that if I walk 
in a certain direction, I shall reach water that I can drink, and 
in this I am deceived ; but my visual percept may be exactly what 
it would be if there really were water. My physics, if adequate, 
must explain not only that there is no water, but also why there 



seems to be water. A mistaken perception is mistaken, not as to 
the percept itself, but as to its causal correlates and antecedents 
and consequents; frequently the mistake is in an animal inference. 
The fact that animal inferences may be mistaken is one reason for 
classifying them as inferences. 

The relation of physical laws to experience is not altogether 
simple. Broadly speaking, laws can be disproved by experience, 
but not proved by it. That is to say, they assert more than ex- 
perience alone would warrant. In the case of the mirage, if I have 
believed it real, and have also assumed that a large lake will not 
dry up in a few hours, I can discover that the mirage caused me to 
have a false belief. But the false belief may have been the belief 
that the lake could not dry up so quickly. The belief in the per- 
sistence of material objects throughout the interval between two 
occasions when they are observed is one which, as a matter of 
logic, cannot be proved by observation. Suppose I were to set up 
the hypothesis that tables, whenever no one is looking, turn into 
kangaroos ; this would make the laws of physics very complicated, 
but no observation could refute it. The laws of physics, in the 
form in which we accept them, must not only be in agreement 
with observation, but must, as regards what is not observed, have 
certain characteristics of simplicity and continuity which are not 
empirically demonstrable. In general, we think that physical 
phenomena are not affected by being observed, although this is 
not thought to be strictly true as regards the minute phenomena 
upon which quantum theory is based. 

Physics, assuming it perfected, would have two characteristics. 
In the first place, it would be able to predict percepts ; no percep- 
tion would be contrary to what physics had led us to expect. In 
the second place, it would assume unobserved physical occurrences 
to be governed by causal laws as similar as possible to those that 
we infer from cases of continuous observation. For example, if I 
watch a moving body, the motion that I see is sensibly continuous ; 
I therefore assume that all motion, whether observed or not, is 
approximately continuous. 

This brings us to the question of causal laws and physical 
space-time.(Physical space-time, as we have seen, is an inference 
from perceptual space and time; it contains all observed occur- 
rences, and also all unobserved occurrences. But since it is in- 
ferential, the location of an occurrence in it is also inferential. 



The locating of events in physical space-time is effected by two 
methods. First, there is a correlation between perceptual space 
and time and physical space-time, though this correlation is only 
rough and approximate. Second, the causal laws of physics assign 
an order to the events concerned, and it is partly by means of them 
that unobserved events are located in space- time .S 

A causal law, as I use the term, is any law which, if true, makes 
it possible, given a certain number of events, to infer something 
about one or more other events. For example, “planets move in 
ellipses” is a causal law. If this law is true, since five points de- 
termine an ellipse, five data (theoretically) should enable us to 
calculate the orbit of the planet. Most laws, however, have not 
this simplicity ; they are usually expressed in differential equations. 
When they are so expressed they assume an order: each event 
must have four co-ordinates, and neighbouring events are those 
whose co-ordinates are very nearly the same. But the question 
arises : how do we assign co-ordinates to events in physical space- 
time? I maintain that, in doing so, we make use of causal laws. 
That is to say, the relation of causal laws to space-time order is a 
reciprocal one. The correct statement is: Events can be arranged 
in a four-dimensional order such that, when so arranged, they 
are interconnected by causal laws which are approximately 
continuous, i.e. events whose co-ordinates differ very little also 
differ very little. Or rather: Given any event, there is a series of 
closely similar events, in which the time- co-ordinate varies 
continuously from rather less to rather more than that of the 
given event, and in which the space-co-ordinates vary continuously 
about those of the given event. This principle, apparently, does 
not hold for quantum transitions, but it holds for macroscopic 
events, and for all events (such as light-waves) where there is no 

The correlation between physical and perceptual space-time, 
which is only approximate, proceeds as follows. In visual space, 
if objects are near enough for differences of depth to be per- 
ceptible, every visual percept has three polar co-ordinates, which 
may be called distance, up-and-downness, and right-and-leftness. 
All these are qualities of the percept, and all are measurable. We 
may assign the same numerical co-ordinates to the physical 
object which we are said to be seeing, but these co-ordinates no 
longer have the same meaning as they have in visual space. It is 



because they do not have the same meaning that it is possible for 
the correlation to be only rough — for example, if the object is seen 
through a refracting medium. But although the correlation is 
rough, it is very useful in establishing a first approximation to the 
co-ordinates of events in physical space-time. The subsequent 
corrections are effected by means of causal laws, of which the 
refraction of light may again serve as an example. 

There is no logical reason why there should be such causal 
laws, or a known relation establishing such a four-dimensional 
order among events. The usual argument for the acceptance of 
physical laws is that they are the simplest hypotheses hitherto 
devised that are consistent with observation wherever observation 
is possible. They are not, however, the only hypotheses consistent 
with observation. Nor is it clear by what right we objectify our 
preference for simple laws. 

What physics says about the world is much more abstract than 
it seems to be, because we imagine that its space is what we know 
in our own experience, and that its matter is the kind of thing that 
feels hard when we touch it. In fact, even assuming physics true, 
what we know about the physical world is very little. Let us first 
consider theoretical physics in the abstract, and then in relation 
to experience. 

As an abstract system, physics, at present, says something like 
this: there is a manifold, called the manifold of events, which has 
a system of relations among its terms by means of which it acquires 
a certain four-dimensional geometry. There is an extra-geo- 
metrical quantity called “energy”, which is unevenly distributed 
throughout the manifold, but of which some finite amount exists 
in every finite volume. The total of energy is constant. The laws 
of physics are laws as to the changes in the distribution of energy. 
To state these laws, we have to distinguish two kinds of regions, 
those that are called “empty”, and those that are said to contain 
“matter”. There are very small material systems called “atoms”; 
each atom may contain any one of a certain discrete denumerable 
series of amounts of energy. Sometimes it suddenly parts with a 
finite amount of energy to the non-material environment, some- 
times it suddenly absorbs a finite amount from the environment. 
The laws as to these transitions from one energy level to another 
are only statistical. In a given period of time, if not too short, 
there will be, in a given state of the environment, a calculable 


human knowledge: its scope and limits 

number of transitions of each possible kind, the smaller transitions 
being commoner than the greater. 

In “empty space” the laws are simpler and more definite. 
Parcels of energy that leave an atom spread outward equally in all 
directions, travelling with the velocity of light. Whether a parcel 
travels in waves or in little units or in something which is a com- 
bination of both, is a matter of convention. Everything proceeds 
simply until the radiant energy hits an atom, and then the atom 
may absorb a finite amount of it, with the same individual inde- 
terminancy and statistical regularity as applies to the emission of 
energy by atoms. 

The amount of energy emitted by an atom in a given transition 
determines the “frequency” of the radiant energy that results. 
And this in turn determines the kinds of effects that the radiant 
energy can have upon any matter that it may encounter. “Fre- 
quency” is a word associated with waves, but if the wave theory 
of light is discarded “frequency” may be taken as a measurable 
but undetermined quality of a radiation. It is measurable by its 

So much for theoretical physics as an abstract logical system. 
It remains to consider how it is connected with experience. 

Let us begin with the geometry of space-time. We assume that 
the position of a point in space-time can be determined by four 
real numbers, called co-ordinates; it is also generally supposed, 
though this is not essential, that to every set of four real numbers 
as co-ordinates (if not too great) a position in space-time corre- 
sponds. It will simplify exposition to adopt this supposition. If 
we do, the number of positions in space-time is the same as the 
number of real numbers, which is called c. Now of every class of c 
entities we can assert every kind of geometry in which there is a 
one-one correspondence between a position and a finite ordered 
set of real numbers (co-ordinates). Therefore to specify the 
geometry of a manifold tells us nothing unless the ordering 
relation is given. Since physics is intended to give empirical 
truth, the ordering relation must not be a purely logical one, such 
as might be constructed in pure mathematics, but must be a 
relation defined in terms derived from experience. If the ordering 
relation is derived from experience, the statement that space- time 
has such-and-such a geometry is one having a substantial empirical 
content, but if not, not. 



I suggest that the ordering relation is contiguity or compresence, 
in the sense in which we know these in sensible experience. Some- 
thing must be said about these. 

Contiguity is a property given in sight and touch. Two portions 
of the visual field are contiguous if their apparent distances and 
their angular co-ordinates (up-and-down, right-and-left) differ 
very little. Two parts of my body are contiguous if the qualities 
by which I locate a touch in the two parts differ very little. Conti- 
guity is quantitative, and therefore enables us to make series of 
percepts: if A and B and C are contiguous, but B is more conti- 
guous to both A and C than they are to each other, B is to be put 
between A and C. There is also contiguity in time. When we 
hear a sentence, the first and second words are more contiguous 
than the first and third words. In this way, by means of spatial 
and temporal contiguity, our experiences can be arranged in an 
ordered manifold. We may assume that this ordered manifold is 
a part of the ordered manifold of physical events, and is ordered 
by the same relation. 

For my part, however, I prefer the relation of “compresence”. 
If we use this relation, we suppose that every event occupies a 
finite amount of space-time, that is to say, no event is confined 
to a point of space or an instant of time. Two events are said to be 
“compresent” when they overlap in space-time; this is the defini- 
tion for abstract physics. But we need, as we saw, a definition 
derived from experience. As an ostensive definition from exper- 
ience I should give the following: Two events are “compresent” 
when they are related in the way in which two simultaneous parts 
of one experience are related. At any given moment, I am seeing 
certain things, hearing others, touching others, remembering 
others, and expecting yet others. All these percepts, recollections, 
and expectations are happening to me now ; I shall say that they 
are mutually “compresent”. I assume that this relation, which 
I know in my own experience, can also hold between events that 
are not experienced, and can be the relation by which space-time 
order is constructed. This will have as a consequence that two 
events are compresent when they overlap in space- time, which, 
if space-time order is taken as already determined, may serve, 
within physics , as the definition of compresence. 

Compresence is not the same thing as simultaneity, though it 
implies it. Compresence, as I mean it, is to be taken as known 


HUMAN knowledge: its scope and limits 

through experience, and having only an ostensive definition. Nor 
should I define “compresence” as “ simultaneity in one person's 
experience”. I should object to this definition on two grounds: 
first, that it could not be extended to physical occurrences ex- 
perienced by no one; second, that “experience” is a vague word. 
I should say that an event is “experienced” when it gives rise to 
a habit, and that broadly speaking this only happens if the event 
occurs where there is living matter. If this is correct, “experience” 
is not a fundamental concept. 

The question now arises: can we construct space-time order 
out of compresence alone, or do we need something further? Let 
us take a simplified hypothesis. Suppose there are n events, a l9 
a 2 , . . . a n9 and suppose a x is compresent only with a 2 , a 2 is 
compresent with a x and a 3 , a 3 , with a 2 and a 4 , and so on. We can 
then construct the order a v a 2l . . . a n . We shall say that an event 
is “between” two others if it is compresent with both, but they 
are not compresent with each other; and, more generally, if 
a t b 9 c are three different events, we shall say that b is “between” 
a and c if the events compresent with both a and c are a proper 
part of the events compresent with b . This may be taken as the 
definition of “between”. Supplemented by suitable axioms, it 
will generate the kind of order we want. 

It should be observed that we cannot construct space-time 
order out of Einstein's relation of “interval”. The interval between 
two parts of a light- ray is zero, and yet we have to distinguish 
between a light-ray that goes from A to B and one that goes 
from B to A. This shows that “interval” alone does not 

If the above point of view is adopted, points in space-time 
become classes of events. I have dealt with this subject in “The 
Analysis of Matter”, and in Chapters VI and VIII of this Part, 
and will therefore say no more about it. 

So much for the definition of space-time order in terms of 
experience. It remains to restate the connection of physical 
events in the outer world with percepts. 

When energy emitted by matter as a result of quantum transi- 
tions travels, without further quantum transitions, to a given 
part of a human body, it sets up a train of quantum transitions 
which ultimately reach the brain. Assuming the maxim “same 
cause, same effect”, with its consequence, “different effects, 



different causes”, it follows that, if two trains of radiant energy, 
falling on the same point of the body, cause different percepts, 
there must be differences in the two trains, and therefore in the 
quantum transitions that gave rise to them. Assuming the existence 
of causal laws, this argument seems unobjectionable, and gives a 
basis for the inference from perceptions to the material source 
of the process by which they are caused. 

I think — though I say this with hesitation — that the distinction 
between spatial and temporal distance requires the consideration 
of causal laws. That is to say, if there is a causal law connecting 
an event A with an event B, then A and B are separated in time, 
and it is a matter of convention whether we shall also consider 
them separated in space. There are, however, some difficulties 
about this view. A number of people may hear or see something 
simultaneously, and in this case there is a causal connection with 
no time- interval. But in such a case the connection is indirect, 
like that connecting brothers or cousins ; that is to say, it travels 
first from effect to cause and then from cause to effect. But how 
are we to distinguish cause from effect before we have established 
the time-order ? Eddington says we do so by means of the second 
law of thermodynamics. In a spherical radiation, we take it that 
it travels from a centre, not to it. But I, since I wish to connect 
physics with experience, should prefer to say that we establish 
time-order by means of memory and our immediate experience 
of temporal succession. What is remembered is, by definition, 
in the past; and there are earlier and later within the specious 
present. Anything compresent with something remembered, but 
not with my present experience, is also in the past. From this 
starting-point we can extend the definition of time-order, and the 
distinction of past and future, step by step to all events. We can 
then distinguish cause from effect, and say that causes are always 
earlier than effects. 

According to the above theory, there are certain elements that 
are carried over unchanged from the world of sense to the world 
of physics. These are: the relation of compresence, the relation 
of earlier and later, some elements of structure, and differences in 
certain circumstances — i.e. when we experience different sensa- 
tions belonging to the same sense, we may assume' that their 
causes differ. This is the residue of naive realism that survives in 
physics. It survives primarily because there is no positive argu- 


HUMAN knowledge: its scope and limits 

ment against it, because the resulting physics fits the known 
facts, and because prejudice causes us to cling to naive realism 
wherever it cannot be disproved. Whether there are any 
better reasons than these for accepting physics remains to be 

35 ° 




I T is generally recognized that the inferences of science and 
common sense differ from those of deductive logic and 
mathematics in a very important respect, namely, that, when 
the premisses are true and the reasoning correct, the conclusion 
is only probable . We have reasons for believing that the sun will 
rise to-morrow, and everybody is agreed that, in practice, we can 
behave as if these reasons justified certainty. But when we examine 
them we find that they leave some room, however little, for doubt. 
The doubt that is justified is of three sorts. As regards the first 
two : on the one hand there may be relevant facts of which we are 
ignorant; on the other hand, the laws that we have to assume 
in order to predict the future may be untrue. The former reason 
for doubt does not much concern us in our present inquiry, but 
the latter is one which demands detailed investigation. But there 
is a third kind of doubt, which arises when we know a law to 
the effect that something happens usually, or perhaps in an over- 
whelming majority of instances, though not always; in this case 
we have a right to expect what is usual, though not with 
complete confidence. For example, if a man is throwing dice, 
it very seldom happens that he throws double sixes ten times 
running, although this is not impossible; we have therefore a 
right to expect that he will not do so, but our expectation ought 
to be tinged with doubt. All these kinds of doubt involve some- 
thing that may be called “probability”, but this word is capable 
of different meanings, which it will be important to us to dis- 

Mathematical probability arises always from a combination of 
two propositions, of which one may be completely known, while 
the other is completely unknown. If I draw a card from a pack, 
what is the chance that it will be an ace ? I know completely the 
constitution of a pack of cards, and I am aware that one card 
in thirteen is an ace ; but I am completely ignorant as to which 
card I shall draw. But if I say “probably Zoroaster existed”, I 
am saying something about the degree of uncertainty, or of 
credibility, attaching to the one proposition “Zoroaster existed”. 
This is quite a different concept from that of mathematical 
probability, although in many cases the two are correlated. 



human knowledge: its scope and limits 

Science is concerned to infer laws from particular facts. An 
inference of this sort cannot be deductive, unless, in addition to 
particular facts, there are general laws among our premisses; as 
a matter of pure logic this is fairly evident. It is sometimes thought 
that, though particular facts cannot make a general law certain , 
they can make it probable . Particular facts can certainly cause belief 
in a general proposition; it is our experience of particular men 
dying that has caused us to believe that all men are mortal. But 
if we are justified in believing that all men are mortal, that must 
be because, as a general principle, certain kinds of particular facts 
are evidence of general laws. And since deductive logic knows 
no such principle, any principle which will justify inference from 
the particular to the general must be a law of nature, i.e. a state- 
ment that the actual universe has a certain character which it 
would be possible for it not to have. I shall undertake the search 
for some such principle or principles in Part VI ; in Part V I shall 
only contend that induction by simple enumeration is not such 
a principle, and unless severely restricted is demonstrably invalid. 

We infer, in science, not only laws, but also particular facts. 
If we read in the newspaper that the King is dead, we infer that 
he is dead; if we find that we shall have to make a long railway 
journey without a chance of a meal, we infer that we shall be 
hungry. All such inferences can only be justified if it is possible 
to ascertain laws. If there were not general laws, every man’s 
knowledge would be confined to what he himself has experienced. 
It is more necessary that there should be laws than that they 
should be known. If A is always followed by B, and an animal, 
seeing A, expects B, the animal may be said to know that B is 
coming without having knowledge of the general law. But although 
some knowledge of facts as yet unperceived can be acquired in 
this way, it is impossible to get far without knowledge of general 
laws. Such laws, in general, state probabilities (in one sense), and 
are themselves only probable (in another). E.g. it is probable 
(in one sense) that if you have cancer it is probable (in 
another sense) that you will die. This state of affairs makes 
it evident that we cannot understand scientific method without 
a previous investigation of the different kinds of probability. 

Although such an investigation is necessary, I do not think 
that probability has quite the importance attached to it by some 
authors. The importance that it has arises in two ways. On the 



one hand, we need, among the premisses of science, not only 
data derived from perception and memory, but also certain 
principles of synthetic inference, which cannot be established 
by deductive logic or by arguments from experience, since they 
are presupposed in all inference from experienced facts to other 
facts or to laws. These premisses may be admitted to be in some 
degree uncertain, i.e. to have not the highest “degree of credi- 
bility”. It will be part of our analysis of this form of probability 
to maintain, in spite of Keynes's adverse opinion, that data and 
inferential premisses may be uncertain. This is one way in which 
the theory of probability is relevant, but there is also another. 
It appears that we frequently know (in some sense of the word 
“know") that something happens usually, but perhaps not always 
— e.g. that lightning is followed by thunder. In that case, we have 
a class of cases A, of which we have reason to believe that most 
belong to the class B. (In our illustration, A is times shortly after 
lightning, and B is times when thunder is audible.) In such 
circumstances, given an instance of the class A concerning which 
we do not know whether it is an instance of the class B, we have 
a right to say that it is “probably” a member of the class B. Here 
“probably” has not the meaning that it has when we are speaking 
of degrees of credibility, but the quite different meaning that it 
has in the mathematical theory of probability. 

For these reasons, and also because probability- logic is much 
less complete and incontrovertible than elementary logic, it is 
necessary to develop the theory of probability in some detail, and 
to examine various controversial questions of interpretation. It 
is to be remembered that the whole discussion of probability is 
of the nature of prolegomena to the investigation of the postulates 
of scientific inference. 


Chapter I 


attempts to establish a logic of probability have been 
L\ numerous, but to most of them there have been fatal 
L ^objections. One of the causes of faulty theories has been 
failure to distinguish — or rather, determination to confound — 
essentially different concepts which, so far as usage goes, have 
an equal right to be called “probability”. I propose, in this 
Chapter, to make a preliminary exploration of these different 
concepts in a discursive manner, leaving to later Chapters the 
attempt to reach precise definitions. 

The first large fact of which we have to take account is the 
existence of the mathematical theory of probability. There is, 
among mathematicians who have concerned themselves with this 
theory, a fairly complete agreement as to everything that can be 
expressed in mathematical symbols, but an entire absence of 
agreement as to the interpretation of the mathematical formulae. 
In such circumstances, the simplest course is to enumerate the 
axioms from which the theory can be deduced, and to decide that 
any concept which satisfies these axioms has an equal right, from 
the mathematician’s point of view, to be called “probability”. If 
there are many such concepts, and if we are determined to choose 
between them, the motives of our choice must lie outside mathe- 

There is one very simple concept which satisfies the axioms 
of the theory of probability, and which is on other accounts 
advantageous. Given a finite class B which has n members, and 
given that m of these belong to some other class A, then we say 
that, if a member of B is chosen at random, the chance that it 
will belong to the class A is m/w. Whether this definition is 
adequate to the uses that we wish to make of the mathematical 
theory of probability, is a question which we shall have to inves- 
tigate at a later stage; if it is not, we shall have to look for some 
other interpretation of mathematical probability. 

It must be understood that there is here no question of truth 
or falsehood. Any concept which satisfies the axioms may be 
taken to be mathematical probability. In fact, it might be desirable 



to adopt one interpretation in one context, and another in another, 
for convenience is the only guiding motive. This is the usual 
situation in the interpretation of a mathematical theory. For 
example, as we have seen, all arithmetic can be deduced from 
five axioms enumerated by Peano, and therefore, if all we want 
of numbers is that they should obey the rules of arithmetic, we 
may define as the series of natural numbers any series satisfying 
Peano’s five axioms. Now these axioms are satisfied by any pro- 
gression, and, in particular, by the series of natural numbers 
starting, not with o, but with ioo, or 1,000, or any other finite 
integer. It is only if we decide that we want our numbers to serve 
for enumeration, not only for arithmetic, that we have a motive 
for choosing the series that starts with o. Similarly, in the case 
of the mathematical theory of probability, the interpretation to 
be chosen may depend upon the purpose we have in view. 

The word “probability” is often used in ways that are not, or 
at least not obviously, capable of interpretation as the ratio of 
the numbers of two finite classes. We may say : Probably Zoroaster 
existed, probably Einstein’s theory of gravitation is better than 
Newton’s, probably all men are mortal . 1 In these cases, we might 
maintain that there is evidence of a certain kind, which is known 
to be conjoined with a conclusion of a certain kind in a large 
majority of cases; in this way, theoretically, the definition of 
probability as the ratio of the numbers of two classes might 
become applicable. It is possible, therefore, that instances such 
as the above do not involve a new meaning of “probability”. 

There are, however, two dicta which we are all inclined to 
accept without much examination, but which, if accepted, involve 
an interpretation of “probability” which it seems impossible to 
reconcile with the above definition. The first of these dicta is 
Bishop Butler’s maxim that “probability is the guide of life”. The 
second is the maxim that all our knowledge is only probable, 
which has been specially emphasized by Reichenbach. 

Bishop Butler’s maxim is obviously valid according to one very 
common interpretation of “probability”. When — as is usually the 
case — I am not certain what is going to happen, but I must act 
upon one or other hypothesis, I am generally well advised to 
choose the most probable hypothesis, and I am always well 
advised to take account of probability in making my decision. 

1 Not to be confused with “all men are probably mortal”. 


HUMAN knowledge: its scope and limits 

But there is an important logical difference between this kind 
of probability and the mathematical kind, namely that the latter 
is concerned with propositional functions 1 and the former with 
propositions. When I say that the chance of a coin coming heads 
is a half, that is a relation between the two propositional functions 
“ x is a toss of the coin” and “ x is a toss of the coin which comes 
heads”. If I am to infer that, in a particular case, the chance 
of heads is a half, I must state that I am considering the particular 
case solely as an instance. If I could consider it in all its par- 
ticularity, I should, in theory, be able to decide whether it will 
fall heads or fall tails, and I should no longer be in the domain 
of probability. When we use probability as a guide to conduct, 
it is because our knowledge is inadequate; we know that the 
event in question is one of a class B of events, and we may know 
what proportion of this class belongs to some class A in which 
we are interested. But the proportion will vary according to our 
choice of the class B ; we shall thus obtain different probabilities, 
all equally valid from the mathematical standpoint. If probability 
is to be a guide in practice, we must have some way of selecting 
one probability as the probability. If we cannot do this, all the 
different probabilities remain equally valid, and we shall be left 
without guidance. 

Let us take an instance, in which every sensible man is guided 
by probability, I mean life insurance. I ascertain the terms on 
which some company is willing to insure my life, and I have to 
decide whether insurance on these terms is likely to prove an 
advantageous bargain, not to insurers in general, but to me. 

My problem is different from that of the insurance company, 
and much more difficult. The insurance company is not interested 
in my individual case: it offers insurance to all members of a 
certain class, and need only take account of statistical averages. 
But I may believe that I have special reasons to expect a long 
life, or that I am like the Scotchman who died the day after 
completing his insurance, remarking with his last breath “I always 
was a lucky fellow”. Every circumstance of my health and my 
way of life is relevant, but some of these may be so uncommon 
that I can get no reliable help from statistics. At last I decide to 

1 I.e. with sentences containing an undefined variable — e.g. “A is a 
man” — which become propositions when we assign a value to the 
variable — in the above case “A”. 



consult a medical man, who, after a few questions, remarks 
genially: “Oh, I expect you’ll live to be 90”. I am painfully aware, 
not only that his judgment is slap-dash and unscientific, but also 
that he wishes to please me. The probability at which I finally 
arrive is thus something quite vague and quite incapable of 
numerical measurement; but it is upon this vague probability 
that, as a disciple of Bishop Butler, I have to act. 

The probability which is the guide of life is not the mathe- 
matical kind, not only because it is not relative to arbitrary data, 
but to all data that bear on the question at issue, but also because 
it has to take account of something which lies wholly outside the 
province of mathematical probability, and which may be called 
“intrinsic doubtfulness”. This is what is relevant when it is said 
that all our knowledge is only probable. Consider for example 
a distant memory which has grown so dim that we can no longer 
trust it with any confidence, a star so faint that we are not sure 
whether we really see it, or a noise so slight that we think it may 
be only imagined. These are extreme cases, but in a lesser degree 
the same sort of doubtfulness is very common. If we assert, as 
Reichenbach does, that all our knowledge is doubtful, we cannot 
define this doubtfulness in the mathematical way, for in the com- 
piling of statistics it is assumed that we know whether or not 
this A is a B, e.g. whether this insured person has died. Statistics 
are built up on a structure of assumed certainty as to past 
instances, and a doubtfulness which is universal cannot be merely 

I think, therefore, that everything we feel inclined to believe 
has a “degree of doubtfulness”, or, inversely, a “degree of 
credibility”. Sometimes this is connected with mathematical 
probability, sometimes not; it is a wider and vaguer conception. 
It is not, however, purely subjective. There is a cognate subjective 
conception, namely, the degree of conviction that a man feels 
about any of his beliefs, but “credibility”, as I mean it, is objective 
in the sense that it is the degree of credence that a rational man 
will give. When I add up my accounts, I give some credence to 
the result the first time, considerably more if I get the same result 
the second time, and almost full conviction if I get it a third time. 
This increase of conviction goes with an increase of evidence, 
and is therefore rational. In relation to any proposition about 
which there is evidence, however inadequate, there is a cor- 


human knowledge: its scope and limits 

responding “degree of credibility”, which is the same as the 
degree of credence given by a man who is rational. (This latter 
may perhaps be regarded as a definition of the word “rational”.) 
The importance of probability in practice is due to its connec- 
tion with credibility, but if we imagine this connection to be 
closer than it is, we bring confusion into the theory of 

The connection between credibility and subjective conviction 
is one that can be studied empirically; we need not, therefore, 
have any views on this subject in advance of the evidence. A 
conjurer, for instance, can arrange circumstances in a manner 
known to himself but calculated to deceive his audience ; he can 
thus acquire data as to how to cause untrue convictions, which 
are likely to be useful in advertising and propaganda. We cannot 
so easily study the relation of credibility to truth, because we 
commonly accept a high degree of credibility as sufficient evidence 
of truth, and if we do not do so we can no longer discover any 
truths. But we can discover whether propositions having high 
credibility form a mutually consistent set, since the set contains 
the propositions of logic. 

I suggest, as a result of the above preliminary discussion, that 
two different concepts each, on the basis of usage, have an equal 
claim to be called “probability”. The first of these is mathematical 
probability, which is numerically measurable and satisfies the 
axioms of the probability- calculus ; this is the sort that is involved 
in the use of statistics, whether in physics, in biology, or in the 
social sciences, and is also the sort that we hope is involved in 
induction. This sort of probability has to do always with classes, 
not with single cases except when they can be considered merely 
as instances. 

But there is another sort, which I call “degree of credibility”. 
This sort applies to single propositions, and takes account always 
of all relevant evidence. It applies even in certain cases in which 
there is no known evidence. The highest degree of credibility 
to which we can attain applies to most perceptive judgments; 
varying degrees apply to judgments of memory, according to their 
vividness and recentness. In some cases the degree of credibility 
can be inferred from mathematical probability, in others it 
cannot; but even when it can it is important to remember that 
it is a different concept. It is this sort, and not mathematical 



probability, that is relevant when it is said that all our know- 
ledge is only probable, and that probability is the guide of 

Both kinds of probability demand discussion. I shall begin with 
mathematical probability. 


Chapter II 


I N this Chapter I propose to treat the theory of probability 
as a branch of pure mathematics, in which we deduce the 
consequences of certain axioms without seeking to assign this 
or that interpretation 1 to them. It is to be observed that, while 
interpretation, in this field, is controversial, the mathematical 
calculus itself commands the same measure of agreement as any 
other branch of mathematics. This situation is in no way peculiar. 
The interpretation of the infinitesimal calculus was for nearly 
two hundred years a matter as to which mathematicians and 
philosophers debated; Leibniz held that it involved actual infini- 
tesimals, and it was not till Weierstrass that this view was definitely 
disproved. To take an even more fundamental example: there has 
never been any dispute as to elementary arithmetic, and yet the 
definition of the natural numbers is still a matter of controversy. 
We need not be surprised, therefore, that there is doubt as to 
the definition of probability though there is none (or very little) 
as to the calculus of probability. 

Following Johnson and Keynes, we will denote by “ pjh ” the 
undefined notion: “The probability of p given A”. When I say 
that this notion is undefined, I mean that it is only defined by 
the axioms or postulates about to be enumerated. Anything satis- 
fying these axioms is an “interpretation” of the calculus of 
probability, and it is to be expected that there will be many 
possible interpretations. No one of these is more correct or more 
legitimate than another, but some may be more important than 
others. So, in finding an interpretation of Peano’s five axioms for 
arithmetic, the interpretation in which the first number is o is 
more important than that in which it is 3781 ; it is more important 
because it enables us to identify the interpretation of the formalist 
conception with the conception recognized in enumeration. But 
for the present we will ignore all questions of interpretation, and 
proceed with the purely formal treatment of probability. 

The axioms or postulates required are given in much the same 

1 As to “interpretation”, see Part IV, Chapter I. 


way by different authors. The following statement is taken from 
Professor C. D. Broad. 1 The axioms are: 

I. Given p and A, there is only one value of />/A. We can 
therefore speak of “ the probability of p given A”. 

II. The possible values of p/h are all the real numbers from 
o to i, both included. (In some interpretations we confine the 
possible values to rational numbers; this is a question I shall 
discuss later.) 

III. If A implies p, then />/A = i. (We use “i” to denote 

IV. If A implies not-/), then pjh = o. (We use “o” to denote 

V. The probability of both p and q given A is the probability 
of p given A multiplied by the probability of q given p and A, 
and is also the probability of q given A multiplied by the probability 
of p given q and A. 

This is called the “conjunctive” axiom. 

VI. The probability of p and/or q given A is the probability 
of p given A plus the probability of q given A minus the probability 
of both p and q given A. 

This is called the “disjunctive” axiom. 

It is immaterial, for our purposes, whether these axioms are 
all necessary ; what concerns us is only that they are sufficient. 

Some observations are called for as regards these axioms. It 
is obvious that II, III, and IV embody, in part, conventions 
which might easily be changed. If, when they are adopted, the 
measure of a given probability is x , we might equally well adopt 
as its measure any number / (#) which increases as x increases ; 
for i and o in III and IV we should then substitute /(i) and / (o). 

According to the above axioms, a proposition which must be 
true if the data are true is to have the probability i in relation 
to the data, and one which must be false if the data are true is 
to have the probability o in relation to the data. 

It is important to observe that our fundamental concept p/h 
is a relation of two propositions (or conjunctions of propositions), 
not a property of a single proposition p. This distinguishes 
probability as it occurs in the mathematical calculus from proba- 
bility as required as a guide to practice, for the latter kind has 
to belong to a proposition in its own right, or, at least, in relation 
1 Mind, N.S. No. 210, p. 98. 



to data which are not arbitrary, but determined by the problem 
and the nature of our knowledge. In the calculus, on the contrary, 
the choice of the data A is wholly arbitrary. 

Axiom V is the “conjunctive” axiom. It gives the chance that 
each of two events will happen. For example: if I draw two cards 
from a pack, what is the chance that they are both red? Here 
“A” represents the datum that the pack consists of 26 red cards 
and 26 black ones; “p” stands for the statement “the first card 
is red” and “5” for the statement “the second card is red”. Then 
“(p and #)/A” is the chance that both are red, p/h is the chance 
that the first is red, “<7/ ( p and A)” is the chance that the second is 
red, given that the first is red. Obviously p /A = 1 /2, q/(p and A) = 
25/51. Thus by the axiom the chance that both are red is \ • ff . 

Axiom VI is the “disjunctive” axiom. In the above illustration, 
it gives the chance that at least one of the cards is red. It says 
that the chance that at least one is red is the chance that the first 
is red plus the chance that the second is red (when it is not given 
whether the first is red or not) minus the chance that both are red. 
This is £ + i — £ * 5 1 » using the result obtained above by the 
use of the conjunctive axiom. 

It is obvious that, by means of Axioms V and VI, given the 
separate probabilities of any finite collection of events, we can 
calculate the probability of their all happening, or of at least one 
of them happening. 

From the conjunctive axiom it follows that 

pl(q and A) = 

(p/h) X ql(p and h) 


This is called the “principle of inverse probability”. Its utility 
may be illustrated as follows. Let p be some general theory, and 
q an experimental datum relevant to p. Then p/h is the probability 
of the theory p on the previously known data, qjh is the probability 
of q on the previously known data, and qj{p and A) is the proba- 
bility of q if p is true. Thus the probability of the theory p after q 
has been ascertained is got by multiplying the previous probability 
of p by the probability of q given p, and dividing by the previous 
probability of q . In the most useful case, the theory p will be 
one which implies 5, so that qj{p and A) = 1. In that case, 

PKS and h) — ^ 



That is to say, the new datum q increases the probability of p 
in proportion to the antecedent improbability of q. In other words, 
if our theory implies something very astonishing, and the astonish- 
ing thing is then found to happen, that greatly increases the 
probability of our theory. 

This principle may be illustrated by the discovery of Neptune 
regarded as a confirmation of the law of gravitation. Here 
p = the law of gravitation, h = all relevant facts known before 
the discovery of Neptune, q = the fact that Neptune was found 
in a certain position. Thus qjh was the antecedent probability 
that a hitherto unknown planet would be found in a certain small 
region of the heavens. Let us take this to be m/w. Then after the 
discovery of Neptune the probability of the law of gravitation 
was njm times as great as before. 

It is obvious that this principle is of great importance in judging 
the bearing of new evidence on the probability of a scientific 
theory. We shall find, however, that it proves somewhat dis- 
appointing, and does not yield such good results as might have 
been hoped. 

There is an important proposition, sometimes called Bayes’s 
theorem, which is as follows. Let p lt p 2 > . . . . p n be n mutually 
exclusive possibilities, of which some one is known to be true; 
let h be the general data, and q some relevant fact. We wish to 
know the probability of one possibility />„ given q y when we know 
the probability of each p r before q was known, and also the 
probability of q given />„ for every r. We have 

p,l(q and h) = [?/(/>, and h) . />,/*] /2[?/(/», and h) . pjh] 


This proposition enables us, for example, to solve the following 
problem : We are given n + i bags, of which the first contains 
n black balls and no white ones, the second contains n — i black 
balls and one white one, the (r + i)** contains n — r black balls 
and r white ones. One bag is chosen, but we do not know which; 
m balls are drawn from it, and are found to be all white; what is 
the probability that the 1 th bag has been chosen? Historically, 
this problem is important in connection with Laplace’s pretended 
proof of induction. 

Take next Bernoulli’s law of large numbers. This states that 
if, on each of a number of occasions, the chance of a certain event 


HUMAN knowledge: its scope and limits 

occurring is p , then, given any two numbers 8 and c, however 
small, the chance that, from a certain number of occasions 
onward, the proportion of occasions on which the event occurs 
will ever differ from p by more than e, is less than 8. 

Let us illustrate by the case of tossing a coin. We suppose that 
heads and tails are equally probable. I say that, in all likelihood, 
after you have tossed often enough, the proportion of heads will 
never again depart from 1/2 by more than c, however small e 
may be; I say further that, however small 8 may be, the chance 
of such a departure anywhere after the n th toss is less than 8, 
provided n is sufficiently great. 

As this proposition is of great importance in the applications 
of probability, for instance to statistics, let us spend a little longer 
in familiarizing ourselves with exactly what it asserts in the above 
case of tossing a coin. I assert first, let us say, that from some 
point onwards, the percentage of heads will always remain between 
49 and 51. You dispute my assertion, and we decide to test it 
empirically as far as this is possible. The theorem then asserts 
that, the longer we go on, the more likely we are to find my 
assertion borne out by the facts, and that, as the number of tosses 
is increased, this likelihood approaches certainty as a limit. You 
are convinced by experiment, we will suppose, that from some 
point onwards the percentage of heads remains always between 
49 and 51, but I now assert that, from some further point 
onwards, it will always remain between 49*9 and 50*1. We 
repeat our experiment, and again you are convinced after a 
time, though probably after a longer time than before. After any 
given number of throws, there is a chance that my assertion may 
not be verified, but this chance diminishes as the number of 
throws increases, and can be made less than any assigned chance, 
however small, by going on long enough. 

The above proposition is easy to deduce from the axioms, but 
cannot, of course, be adequately tested empirically, since it 
involves infinite series. If the tests we can make seem to confirm 
it, the opponent can always say they would not have done so if 
we had gone on longer; and if they seem not to confirm it, the 
supporter of the theorem can equally say that we have not gone 
on long enough. The theorem cannot, therefore, be either proved 
or disproved by empirical evidence. 

The above are the principal propositions in the pure theory 



of probability that are important in our inquiry. I will, however, 
say something more on the subject of the n + i bags, containing 
n balls each, some white and some black, the i ,A bag con- 
taining r white balls and n — r black balls. The data are as follows : 
I know that the bags have these varying numbers of white and 
black balls, but there is no way of distinguishing them from the 
outside. I choose one bag at haphazard, and draw from it, one 
by one, m balls which I do not replace after drawing them. They 
turn out to be all white. In view of this fact, I want to know two 
things: first, what is the chance that I have chosen the bag that 
has only white balls? second, what is the chance that the next 
ball I draw will be white ? 

We proceed as follows. Let h be the fact that the bags are 
constituted in the above manner, and q the fact that m white 
balls have been drawn ; also let p r be the hypothesis that we have 
chosen the bag containing r white balls. It is obvious that r must 
be at least as great as m, i.e. 

If r is less than m , then pjqh = o and qjpji = o. 

After some calculation, it turns out that the chance that we have 

chosen the bag in which all the balls are white is — 

n + i 

We now want to know the chance that the next ball will be 
white. After some further calculation, it turns out that this 

. m + 1 
chance is — — . 

m + 2 

Note that this is independent of «, and that, if m is large, it is 
very nearly i . 

I have not, in the above outline, included any arguments on 
the subject of induction, which I postpone to a later stage. I shall 
first consider the adequacy of a certain interpretation of proba- 
bility, in so far as this can be considered independently of the 
problems connected with induction. 


Chapter III 


I N this Chapter we are concerned with a certain very simple 
interpretation of “probability”. We have, first, to show that 
it satisfies the axioms of Chapter II, and then to consider, 
in a preliminary way, how far it can be made to cover ordinary 
uses of the word “probability”. I shall call it “the finite frequency 
theory”, to distinguish it from another form of frequency theory 
which we shall consider later. 

The finite frequency theory starts from the following definition: 
Let B be any finite class, and A any other class. We want to 
define the chance that a member of B chosen at random will be 
a member of A, e.g. that the first person you meet in the street 
will be called Smith. We define this probability as the number 
of B’s that are A’s divided by the total number of B’s. We denote 
this by the symbol A/B. 

It is obvious that a probability so defined must be a rational 
fraction or o or i . 

A few illustrations will make the purport of this definition clear. 
What is the chance that an integer less than io, chosen at random, 
will be a prime? There are 9 integers less than 10, and 5 of them 
are primes; therefore the chance is 5/9. What is the chance that 
it rained in Cambridge on my birthday last year, assuming that 
you do not know when my birthday is ? If m was the number of 
days on which it rained, the chance is mj 365. What is the chance 
that a man whose name occurs in the London telephone book 
will be called Smith ? To solve this problem, you must first count 
the entries under the name “Smith”, and then count all the entries, 
and divide the former number by the latter. What is the chance 
that a card drawn at random from a pack will be a spade? 
Obviously 13/52, i.e. 1/4. If you have drawn a spade, what is 
the chance that you will draw another? The answer is 12/51. 
In a throw of two dice, what is the chance that the numbers will 
add up to 8 ? There are 36 ways in which the dice may fall, and 
in 5 of these the numbers add up to 8, so the chance is 5/36. 

It is obvious that, in a number of elementary cases, the above 
definition gives results that accord with usage. Let us now inquire 
whether probability, so defined, satisfies the axioms. 



The letters p and q and A, which occur in the axioms, must 
now be taken to stand for classes or propositional functions, not 
for propositions. Instead of “ A implies />” we shall have “A is 
contained in />”; “ p and q” will stand for the common part of 
the two classes p and q , while “ p or q” will be the class of all terms 
that belong to either or both of the two classes p and q . 

Our axioms were : 

I. There is only one value of pjh. This will be true unless A is 
null, in which case pjh = o/o. We shall therefore assume that A 
is not null. 

II. The possible values of pjh are all the real numbers from 
o to i. In our interpretation, they will be only the rational num- 
bers, unless we can find a way of extending our definition to 
infinite classes. This cannot be done simply, since division does 
not yield a unique result when the numbers concerned are infinite. 

III. If A is contained in p , then pjh = i. In this case, the 
common part of A and p is A, therefore the above follows from 
our definition. 

IV. If A is contained in not-/), then pjh = o. This is obvious 
on our definition, since in this case the common part of A and p 
is null. 

V. The conjunctive axiom . — This states, on our interpretation, 
that the proportion of members of A which are members of both 
p and q is the proportion of members of A that are members of 
p multiplied by the proportion of members of p and A that are 
members of q. Suppose the number of members of A is a, the 
number of members common to p and A is A, and the number 
of members common to p and q and A is c . Then the proportion 
of members of A that are members of both p and q is cja\ the 
proportion of members of A that are members of p is bja t and 
the proportion of members of p and A that are members of q is 
cjb. Thus our axiom is verified, since c/a — bja X cjb . 

VI. The disjunctive axiom . This says, on our present inter- 

pretation, keeping the above meanings of a , A, and c , and adding 
that d is the number of members of A that are members of p or q 
or both, while e is the number of members of A that are members 
of j, that , 

d b * e C • J L i 

- = - 4 , i.e. d= A + e — c 

a a a a 

which again is obvious. 



HUMAN knowledge: its scope and limits 

Thus our axioms are satisfied if h is a finite class which is not 
null, except that the possible values of a probability are confined 
to rational fractions. 

It follows that the mathematical theory of probability is valid 
on the above interpretation. 

We have, however, to inquire as to the scope of probability so 
defined, which is, prima facie , much too narrow for the uses that 
we wish to make of probability. 

In the first place, we wish to be able to speak of the chance 
that some definite event will have some characteristic, not only 
of the chance that an unspecified member of a class will have it. 
For example: You have already made a throw with two dice, but 
I have not seen the result. What is, for me, the probability that 
you have thrown double sixes? We want to be able to say that 
it is 1/36, and if our definition does not allow us to say so, it is 
inadequate. In such a case, we should say that we are considering 
an event merely as an instance of a certain class; we should say 
that, if a is considered merely as a member of the class B, the 
chance that it belongs to the class A is A/B. But it is not very 
clear what is meant by “considering a definite event merely as 
a member of a certain class”. What is involved in such a case 
is this: We are given some characteristic of an event which, to 
more complete knowledge than ours, is sufficient to determine it 
uniquely; but relatively to our knowledge, we have no way of 
finding out whether it belongs to the class A, though we do know 
that it belongs to the class B. You, who have thrown the dice, 
know whether the throw belongs to the class of double sixes, but 
I do not know this. My only relevant knowledge is that it is one 
of the 36 possible kinds of throws. Or take the following question: 
What is the chance that the tallest man in the United States lives 
in Iowa? Somebody may know who he is; at any rate there is a 
known method of finding out who he is. If this method has been 
successfully employed, there is a definite answer not involving 
probability, namely, either that he does live in Iowa or that he 
does not. But I have not this knowledge. I can ascertain that the 
population of Iowa is m and that of the United States is n , and 
say that, relative to these data, the probability that he lives in 
Iowa is m/n. Thus when we speak of the probability of a definite 
event having some characteristic, we must always specify the data 
relative to which the probability is to be estimated. 



We may say generally: Given any object a , and given that a 
is a member of the class B, we say that, in relation to this datum, 
the probability that a is an A is A/B as previously defined. This 
conception is useful because we often know enough about some 
object to enable us to define it uniquely, without knowing enough 
to determine whether it has this or that property. “The tallest 
man in the United States” is a definite description, which applies 
to one and only one man, but I do not know what man, and 
therefore for me it is an open question whether he lives in Iowa. 
“The card I am about to draw” is a definite description, and in 
a moment I shall know whether this description applies to a red 
or a black card, but as yet I do not know. It is this very common 
condition of partial ignorance as to definite objects that makes 
it useful to apply probability to definite objects, and not only to 
wholly undefined members of classes. 

Although partial ignorance is what makes the above form of 
probability useful , ignorance is not involved in the concept of 
probability, which would still have the same meaning for omnis- 
cience as for us. Omniscience would know whether a is an A, 
but would still be able to say: Relative to the datum that a is 
a B, the probability that a is an A is A/B. 

In the application of our definition to a definite instance, there 
is a possible ambiguity in certain cases. To make this clear, we 
must use the language of properties rather than classes. Let the 
class A be defined by the property <p and the class B by the 
property ip. Then we say : 

The probability that a has the property cp given that it has the 
property ip is defined as the proportion of things having both the 
properties (p and \p to those having the property tp. We denote 
“ a has the property <p” by “(pa”. But if a occurs more than once 
in “<pa” y there will be an ambiguity. E.g. suppose “<pa” is “a 
commits suicide”, i.e. “a kills a”. This is a value of “x kills x” y 
which is the class of suicides; also of “a kills x” y which is the 
class of persons whom a kills ; also of “x kills a” y which is the 
class of persons who kill a . Thus in defining the probability of 
<pa y if “a” occurs more than once in “<pa” y we must indicate 
which of its occurrences are to be regarded as values of a variable 
and which not. 

It will be found that we can interpret all elementary theorems 
in accordance with the above definition. 


HUMAN knowledge: its scope and limits 

Take, for example, Laplace’s supposed justification of induction : 

There are N + i bags, each containing N balls. 

Of the bags, the r + i th contains r white balls and N — r black 
balls. We have drawn from one bag n balls, all white. What is 
the chance 

(a) that we have chosen the bag in which all are white? 

( b ) that the next ball will be white ? 

Laplace says that (a) is ( n + i) /(N + i) and ( b ) is ( n + i )/(» + 2). 

Let us illustrate by some numerical instances. First: Suppose 
there are 8 balls altogether, of which 4 have been drawn, all 
white. What are the chances (a) that we have chosen the bag 
consisting only of white balls, and ( b ) that the next ball drawn 
will be white? 

Let p 9 represent the hypothesis that we have chosen the bag 
with r white balls. The data exclude p 0 , p x> /> 2 , /> 3 . If we have /> 4 , 
there is only one way in which we can have drawn 4 whites, and 
there remain 4 ways of drawing a black, none of drawing a white. 
If we have p 5l there were 5 ways in which we could have drawn 
4 whites, and for each of these there was 1 way of drawing 
another white, and 3 of drawing a black; thus from p & we get 
a contribution of 5 cases where the next is white and 15 where 
it is black. If we have /> 6 , there were 15 ways of choosing 4 whites, 
and when they had been chosen there remained 2 ways of choosing 
a white and 2 of choosing a black ; thus we get from p Q 30 cases 
of another white and 30 where the next is black. If we have p ly 
there are 35 ways of drawing 4 whites, and after they have been 
drawn there remain 3 ways of drawing a white and one of drawing 
a black; thus we get 105 ways of drawing another white and 35 
of drawing a black. If we have /> 8 , there are 70 ways of drawing 
4 whites, and when they have been drawn there are 4 ways of 
drawing another white and none of drawing a black ; thus we get 
from p s 280 cases of a fifth white and none of a black. Adding, 
we have 5 + 30 + 105 + 280, i.e. 420, cases in which the fifth 
ball is white, and 4 -f 15 + 30 + 35, i.e. 84, cases in which the 
fifth ball is black. Therefore the odds in favour of white are 
420 to 84, i.e. s to 1 ; that is to say, the chance of the fifth ball 
being white is 5/6. 

The chance that we have chosen the bag in which all the balls 
are white is the ratio of the number of ways of choosing 4 white 
balls from this bag to the total number of ways of choosing 4 

37 * 


white balls. The former, we have seen, is 70; the latter is 
1 + S + IS + 3S + 70, i.e. 126. Therefore the chance is 70/126, 
i.e. 5/9. 

Both these results are in accordance with Laplace’s formula. 
To take one more numerical example: suppose there are 10 
balls, of which 5 have been drawn, and have been found to be 
all white. What is the chance of p 10 , i.e. of our having chosen 
the bag with only white balls? And what is the chance that the 
next ball will be white ? 

p 5 possible in 1 way; if p 5> no way of another white, 5 of a black. 

p 6 >> 

„ 6ways;„/> 6 , 1 



> 4 

Pi »» 

„ 21 „ ; 2 ways 



, 3 

Ps »» 

»» 56 >> > uPhi 3 »> 



, 2 

P 9 »» 

,, 126 ,, ; ,,/> 9 , 4 „ 



, 1 

P 10 

»» 252 ,, , , , P 10 > 5 »» 



, 0 

Thus the chance of p 10 is 252/[i + 6 + 21 + 56 + 126 + 252] 
252/462, i.e. 6/1 1. 

The ways in which the next ball can be white are 

6 + 21 X 2 + 56 X 3 + 126 X 4 + 252 X 5, i.e. 1980 
and the ways in which it can be black are 

5 + 4x6 + 3x21 + 2x56+ 126, i.e. 330. 


Therefore the odds in favour of white are 1,980 to 330, i.e. 6 to 1, 
so that the chance of another white is 6/7. This again is in 
accordance with Laplace’s formula. 

Let us now take Bernoulli’s law of large numbers. We may 
illustrate it as follows : suppose we toss a coin n times, and put 1 
for every time it comes heads, 2 for every time it comes tails, 
thus forming a number of n digits. We will suppose every possible 
sequence to come just once. Thus if n = 2, we have the four 
numbers 11, 12, 21, 22; if n = 3, we have the 8 numbers in, 
112, 121, 122, 211, 212, 221, 222; if n = 4, we have 16 numbers, 
mi, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 
2122, 2211, 2212, 2221, 2222; 
and so on. Taking the last of the above lists, we find 
1 number all 1 ’s 

4 numbers with three 1 ’s and one 2 
6 „ „ two Ts „ two 2’s 

4 „ „ one 1 „ three 2’s 

1 number all 2*s. 



These numbers, i, 4, 6, 4, 1, are the coefficients in (a+ 6) 4 . 
It is easy to prove that, for n digits, the corresponding numbers 
are the coefficients in (a + b) n . All that Bernoulli’s theorem 
amounts to is that, if n is large, the sum of the coefficients near 
the middle is very nearly equal to the sum of all the coefficients 
(which is z n ). Thus if we take all possible series of heads and 
tails in a large number of tosses, the immense majority have very 
nearly the same number of both ; the majority and the nearness, 
moreover, increase indefinitely as the number of throws increases. 

Though Bernoulli’s theorem is more general and more precise 
than the above statements with equi-probable alternatives, it is 
to be interpreted, on our present definition of “probability”, in 
a manner analogous to the above. It is a fact that, if we form all 
numbers that consist of 100 digits, each of which is either 1 or 2, 
about a quarter have 49 or 50 or 51 digits that are 1, nearly a half 
have 48 or 49 or 50 or 51 or 52 digits that are 1, more than half 
have from 47 to 53 digits that are 1, and about three-quarters 
from 46 to 54. As the number of digits is increased, the pre- 
ponderance of cases in which i’s and 2’s are nearly evenly 
balanced increases. 

Why this purely logical fact should be regarded as giving us 
good ground for expecting that, when we toss a penny a great 
many times, we shall in fact attain an approximately equal number 
of heads and tails, is a different question, involving laws of nature 
in addition to logical laws. I mention it now only to emphasize 
the fact that I am not at present discussing it. 

I want to lay stress on the fact that, in the above interpretation, 
there is nothing about possibility, and nothing which essentially 
involves ignorance. There is merely a counting of members of 
a class B and determining what proportion of them also belong 
to a class A. 

It is sometimes contended that we need an axiom of equi- 
probability — e.g. to the effect that heads and tails are equally 
probable. If this means that in fact they occur with approximately 
equal frequency, the assumption is not necessary to the mathe- 
matical theory, which, as such, is not concerned with actual 

Let us now consider possible applications of the finite-frequency 
definition to cases of probability which might seem to fall out- 
side it. 



First : in what circumstances can the definition be extended to 
infinite collections? Since we have defined a probability as a 
fraction, and since fractions are meaningless when numerator and 
denominator are infinite, it will only be possible to extend the 
definition when there is some means of proceeding to a limit. 
This requires that the a" s, of which we are to estimate the proba- 
bility of their being V s, should form a series, in fact a progression, 
so that they are given as a lt Og, a 3 , . . . a nf . . . where for every 
finite integer n there is a corresponding a n and vice versa. We can 
then denote by the proportion of a ' s up to a n that belong to b. 
If, as n increases, p n approaches a limit, we can define this limit 
as the probability that an a will be a b } We must, however, 
distinguish the case in which the value of p n oscillates about the 
limit from that in which it approaches the limit from one side 
only. If we repeatedly toss a coin, the number of heads will be 
sometimes more than half the total, sometimes less; thus p n 
oscillates about the limit 1/2. But if we consider the proportion 
of primes up to w, this approaches the limit zero from one side 
only: for any finite n , p n is a definite positive fraction, which, for 
large values of «, is approximately 1 /log n. Now 1 /log n approaches 
zero as n increases indefinitely. Thus the proportion of primes 
approaches zero, but we cannot say “no integers are primes ,, ; 
we may say that the chance of an integer being a prime is infini- 
tesimal, but not zero. Obviously the chance of an integer being 
a prime is greater than that of its being (say) both odd and even, 
although the chance is less than any finite fraction, however small. 
I should say that, when the chance that an a is a b is strictly zero, 
we can infer “no a is a b”, but when the chance is infinitesimal 
we cannot make this inference. 

It is to be observed that, unless we make some assumption 
about the course of nature, we cannot use the method of pro- 
ceeding to the limit when we are dealing with a series which is 
defined empirically. For example, if we toss a given coin repeatedly, 
and find that the number of heads, as we go on, approaches 
continually nearer to the limit 1/2, that does not entitle us to 
assume that this really would be the limit if we could make our 
series infinite. It may be, for example, that, if n is the number 

1 This limit depends upon the order of the a* s, and therefore belongs 
'to them as a series, not as a class. 


HUMAN knowledge: its scope and limits 

of tosses, the proportion of heads does not approximate strictly 
to 1/2 but to 

where N is a number much larger than any that we can reach 
in actual experiments. In that case, our inductions would begin 
to be empirically falsified just as we were thinking they were 
firmly established. Or again, it might happen, with any empirical 
series, that after a time it became utterly lawless, and ceased in 
any sense to approach a limit. If, then, the above extension to 
infinite series is to be used in empirical series, we shall have to 
invoke some kind of inductive axiom. Without this, there is no 
reason for expecting the later parts of such a series to continue 
to exemplify some law which the earlier parts obey. 

In ordinary empirical judgments of probability, such, for 
example, as are contained in the weather forecast, there is a 
mixture of different elements which it is important to separate. 
The simplest hypothesis — unduly simplified for purposes of illus- 
tration — is that some symptom is observed which, in (say) 90 
per cent of the cases in which it has been previously observed, 
has been followed by rain. In that case, if inductive arguments 
were as indubitable as deductive ones, we should say “there is 
a 90 per cent probability of rain”. That is to say, the present 
moment belongs to a certain class (that of moments when the 
symptom in question is present) of which 90 per cent are moments 
preceding rain. This is probability in the mathematical sense 
which we have been considering. But it is not this alone that 
makes us uncertain whether it will rain. We are also uncertain 
as to the validity of the inference; we do not feel sure that the 
symptom in question will, in the future, be followed by rain nine 
times out of ten. And this doubt may be of two kinds, one scientific, 
the other philosophical. We may, while retaining full confidence 
in scientific procedure in general, feel that, in this case, the data 
are too few to warrant an induction, or that not sufficient 
care has been taken to eliminate other circumstances which may 
have also been present and may be more invariable precursors 
of rain. Or, again, the records may be doubtful: they may have 
been rendered nearly indecipherable by rain, or have been made 
by a man who was shortly afterwards certified as insane. Such 
doubts are within scientific procedure, but there are also the 



doubts raised by Hume: is inductive procedure valid, or is it 
merely a habit which makes us comfortable ? Any or all of those 
reasons may make us hesitant about the 90 per cent chance of 
rain which our evidence inclines us to believe in. 

We have, in cases of this sort, a hierarchy of probabilities. The 
primary level is: Probably it will rain. The secondary level is: 
Probably the symptoms I noticed are a sign of probable rain. The 
tertiary level is: Probably certain kinds of events make certain 
future events probable. Of these three levels, the first is that of 
common sense, the second that of science, and the third that of 

In the first stage, we have observed that, hitherto, A has been 
followed, nine times out of ten, by B; in the past, therefore 
A has made B probable in the sense of finite frequency. We 
suppose without reflection, at this stage, that we may expect the 
same thing in the future. 

In the second stage, without questioning the general possibility 
of inferring the future from the past, we realize that such in- 
ferences should be submitted to certain safeguards, such, for 
example, as those of Mill’s four methods. We realize also that 
inductions, even when conducted according to the best rules, are 
not always verified. But I think our procedure can still be brought 
within the scope of the finite frequency theory. We have made 
in the past a number of inductions, some more careful, some 
less so. Of those made by a certain procedure, a proportion p 
have, so far, been verified ; therefore this procedure, hitherto, has 
conferred a probability p upon the inductions that it sanctioned. 
Scientific method consists largely of rules by means of which p 
(as tested by the past results of past inductions) can be made 
to approach nearer to 1 . All this is still within the finite frequency 
theory, but it is now inductions that are the single terms in our 
estimate of frequency. 

That is to say, we have two classes A and B, of which A consists 
of inductions that have been performed in accordance with certain 
rules, and B consists of inductions which experience hitherto has 
confirmed. If n is the number of members of A, and m is the 
number of members common to A and B, then m/n is the chance 
that an induction conducted according to the above rules will 
have, up to the present, led to results which, when they could 
be tested, were found to be true. 


HUMAN knowledge: its scope and limits 

In saying this, we are not using induction; we are merely 
describing a feature of the course of nature so far as it has been 
observed. We have, however, found a criterion of the excellence 
(hitherto) of any suggested rules of scientific procedure, and we 
have found it within the finite frequency theory. The only novelty 
is that our units are now inductions, not single events. The 
inductions are treated as occurrences, and it is only those that 
have actually occurred that are to be regarded as members of 
our class A. 

But as soon as we argue either that an individual induction 
which has hitherto been confirmed will, or will probably, be con- 
firmed in the future, or that rules of procedure, which have given, 
so far, a large proportion of inductions that have been confirmed 
so far, are likely to give a large proportion of confirmed inductions 
in the future, we have passed outside the finite frequency theory, 
since we are dealing with classes of which the numbers are not 
known. The mathematical theory of probability, like all pure 
mathematics, though it gives knowledge, does not (at least in one 
important sense) give anything new ; induction, on the other hand, 
certainly gives something new, and the only doubt is whether 
what it gives is knowledge. 

I do not want, as yet, to examine induction critically; I wish 
only to make clear that it cannot be brought within the scope of 
the finite frequency theory, even by the device of considering 
a particular induction as one of a class of inductions, since tested 
inductions can only supply inductive evidence in favour of a 
hitherto untested induction. If, then, we say that the principle 
which validates induction is “probable”, we must be using the 
word “probable” in a different sense from that of the finite 
frequency theory; the sense in question must, I should say, be 
what we called “degree of credibility”. 

I incline to think that, if induction, or whatever postulate we 
may decide upon as a substitute, is assumed, all precise and 
measurable probabilities can be interpreted as finite frequencies. 
Suppose I say, for example: “There is a high probability that 
Zoroaster existed”. To substantiate this statement, I shall have 
to consider, first, what is the alleged evidence in his case, and 
then to look out for similar evidence which is known to be either 
veridical or misleading. The class upon which the probability 
depends is not the class of prophets, existent and non-existent, 



for by including the non-existent we make the class somewhat 
vague ; nor can it be the class of existent prophets only, since the 
question at issue is whether Zoroaster belongs to this class. We 
shall have to proceed as follows : There is, in the case of Zoroaster, 
evidence belonging to a certain class A; of all the evidences that 
belong to this class and can be tested, we find that a proportion p 
are veridical; we therefore infer, by induction, that there is a 
probability p in favour of the similar evidence in the case of 
Zoroaster. Thus frequency plus induction covers this use of 

Or suppose we say, like Bishop Butler: “It is probable that 
the universe is the result of design on the part of a Creator”. 
Here we start with such subsidiary arguments as that a watch 
implies a watchmaker. There are very many instances of watches 
known to be made by watchmakers, and none of watches known 
to be not made by watchmakers. There is in China a kind of 
marble which sometimes, by accident, produces what appear to 
be pictures made by artists; I have seen the most astonishing 
examples. But this is so rare that, when we see a picture, we are 
justified (assuming induction) in inferring an artist with a very 
high degree of probability. What remains for the episcopal 
logician, as he emphasizes by the title of his book, is to prove 
the analogy; this may be held doubtful, but cannot well be 
brought under the head of mathematical probability. 

So far, therefore, it would seem that doubtfulness and mathe- 
matical probability — the latter in the sense of finite frequency — 
are the only concepts required in addition to laws of nature and 
rules of logic. This conclusion, however, is only provisional. 
Nothing definitive can be said until we have examined certain 
other suggested definitions of “probability”. 


Chapter IV 


T he frequency interpretation of probability, in a form 
different from that of the previous chapter, has been set 
forth in two important books, both by German professors 
who were then in Constantinople. 1 

Reichenbach’s work is a development of that of v. Mises, and 
is in various ways a better statement of the same kind of theory. 
I shall therefore confine myself to Reichenbach. 

After giving the axioms of the probability calculus, Reichenbach 
proceeds to offer an interpretation which seems to be suggested 
by the case of statistical correlations. He supposes two series 
(#i» x & • • • x n » • • •)> (^d ^2* • • • y n y • • •)» an d two classes O and P. 
Some or all of the x's belong to the class O; what interests him 
is the question : how often do the corresponding y’s belong to 
the class P ? 

Suppose, for example, you were investigating the question 
whether a man is predisposed to suicide by having a nagging wife. 
In this case, the x’s are wives, the ys are husbands, the class O 
consists of naggers, and the class P of suicides. Then given that 
a wife belongs to the class O, our question is: how often does 
her husband belong to the class P ? 

Consider the sections of the two series consisting of the first n 
terms of each. Suppose that, among the first n x’s, there are a 
terms belonging to the class O, and suppose that, of these, there 
are b terms such that the corresponding y belongs to the class P. 
(The corresponding y is the one with the same suffix.) Then we 
say that, throughout the section from x x to x ny the “relative 
frequency” of O and P is bja . (If all the tf’s belong to the class O, 
a = n, and the relative frequency is b/n.) We denote this relative 
frequency by “H n ( 0 , P)”. 

We now proceed to define “the probability of P given O” 
which we denote by “W( 0 , P)”. The definition is: 

1 Richard von Mises, Wahrscheinlichkeit , Statistik und Wahrheit y 
2nd ed. Vienna, 1936 (1st ed. 1928); Hans Reichenbach, Wahrschein - 
lichkeitslehre , Leiden, 1935. See also the latter’s Experience and Prediction , 



W(0, P) is the limit of H n (0, P) as n is indefinitely in- 

This definition can be considerably simplified by the use of 
a little mathematical logic. In the first place, it is unnecessary to 
have two series. For both are assumed to be progressions, and there 
is therefore some one-one correlator of their terms. If this 
is S, to say that a certain y belongs to a class P is equivalent 
to saying that the corresponding x belongs to the class of terms 
having the relation S to some one or other of the members 
of P. E.g. let S be the relation of wife to husband; then if y 
is a married man and x is his wife, to say that y is a govern- 
ment official is true if, and only if, x is the wife of a government 

In the second place, there is no advantage in admitting 
the case in which not all the x’s belong to the class O. The 
definition is only appropriate if an infinite number of the x's 
belong to the class O; in that case, those that belong to O form 
a progression, and the rest can be forgotten. Thus we retain 
what is essential in Reichenbach ’s definition if we substitute the 

Let Q be a progression, and a some class, of which, in the 
important cases, there are members, in the series of Q, later than 
any given member. Let m be the number of members of a among 
the first n members of Q. Then W(Q, a) is defined as the limit 
of m/n when n is indefinitely increased. 

Perhaps through inadvertence, Reichenbach speaks as if the 
concept of probability were only applicable to progressions, and 
had no application to finite classes. I cannot think he intends this. 
The human race, for example, is a finite class, and we wish to 
apply probability to vital statistics, which would be impossible 
according to the letter of the definition. As a matter of psycho- 
logical fact, when Reichenbach speaks of the limit for n = infinity, 
he is thinking of the limit as some number which is very nearly 
approached whenever n is large from an empirical point of view, 
i.e. when it is not far short of the maximum that our means of 
observation enable us to reach. He has an axiom or postulate to 
the effect that, when there is such a number for every large 
observable n , it is approximately equal to the limit for n = infinity. 
This is an awkward axiom, not only because it is arbitrary, but 
because most of the series with which we are concerned outside 


HUMAN knowledge: its scope and limits 

pure mathematics are not infinite; indeed it may be doubted 
whether any of them are. We are in the habit of assuming that 
space-time is continuous, which implies the existence of infinite 
series; but this assumption has no basis except mathematical 

I shall assume, in order to make Reichenbach *s theory as 
adequate as possible, that, where finite classes are concerned, 
the definition of the last chapter is to be retained, and that the 
new definition is only intended as an extension enabling us to 
apply probability to infinite classes. Thus his H w (O, P) will be 
a probability, but one applying only to the first n terms of the 

What Reichenbach postulates, as his form of induction, is 
something like this: Suppose we have made N observations as 
to the correlation of O and P, so that we are in a position to 
calculate H n (O, P) for all values of n up to n = N, and suppose 
that, throughout the last half of the values of n , H w (O, P) always 
differs from a certain fraction p by less than c, where e is small. 
Then it shall be posited that, however much we were to increase 
w, H w (O, P) would still lie within these narrow boundaries, and 
therefore W (O, P), which is the limit for n = infinity, will also 
lie within these boundaries. Without this assumption, we can 
have no empirical evidence as to the limit for n = infinity, and 
the probabilities for which the definition is specially designed 
must remain totally unknown. 

In defence of Reichenbach’s theory, in face of the above 
difficulties, two things may be said. In the first place, he may 
contend that it is not necessary to suppose n to approach infinity 
indefinitely; for all practical purposes, it suffices if n is allowed 
to become very large. Suppose, for instance, that we are dealing 
with vital statistics. It does not matter to an insurance company 
what will happen to the statistics if they are prolonged for another 
10,000 years; at most, the next ioo years concern it. If, when 
we have accumulated statistics, we assume that frequencies will 
remain roughly the same until we have ten times as many data 
as we have now, that is enough for almost all practical purposes. 
Reichenbach may say that, when he speaks of infinity, he is using 
a convenient mathematical shorthand, meaning only “a good deal 
more of the series than we have investigated hitherto”. The case 
is exactly analogous, he might say, to that of the empirical deter- 



mination of a velocity. In theory, a velocity can only be deter- 
mined if there is no limit to the smallness of measureable spaces 
and times; in practice, since there is such a limit, the velocity 
at an instant can never be known even approximately. We can, 
it is true, know with a fair measure of accuracy the average 
velocity throughout a short time. But even if we assume a postulate 
of continuity, the average velocity throughout (say) a second gives 
absolutely no indication as to the velocity at a given instant during 
that second. All motion might consist of periods of rest separated 
by instants of infinite velocity. Short of this extreme hypothesis, 
and even if we assume continuity in the mathematical sense, no 
finite velocity at an instant is incompatible with any finite average 
velocity throughout a finite time, however short, which contains 
that instant. For practical purposes, however, this is of no con- 
sequence. Except in a few phenomena such as explosions, if we 
take the velocity at any instant through a very short measurable 
time to be approximately the average velocity during that time, 
the laws of physics are found to be verified. “Velocity at an in- 
stant”, therefore, may be regarded as nothing but a convenient 
mathematical fiction. 

In like manner, Reichenbach may say, when he speaks of the 
limit of a frequency when n is infinite, he means only the actual 
frequency for very large numbers, or rather this frequency with 
a small margin of error. The infinite and the infinitesimal are 
equally unobservable, and therefore (he may say) equally irrelevant 
to empirical science. 

I am inclined to admit the validity of this answer. I only regret 
that I do not find it explicitly in Reichenbach ’s books; I think, 
nevertheless, that he must have had it in mind. 

The second point in favour of his theory is that it applies to 
just the sort of cases in which we wish to use probability argu- 
ments. We wish to use these arguments when we have some data 
as to a certain future event, but not enough to determine its 
character in some respect that interests us. My death, for example, 
is a future event, and if I am insuring my life I may wish to 
know what evidence exists as to the likelihood of death occurring 
in some given year. In such a case we always have a number of 
individual facts recorded in a series, and we assume that the 
frequencies we have found hitherto will more or less continue. 
Or take gambling, from which the whole subject took its rise. 

HUMAN knowledge: its scope and limits 

We are not interested in the mere fact that there are 36 possible 
results of a throw with two dice. What we are interested in is the 
fact (if it be a fact) that in a long series of throws each of these 
36 possibilities will be realized an approximately equal number 
of times. This is a fact which does not follow from the mere 
existence of 36 possibilities. When you meet a stranger, there are 
exactly two possibilities: on the one hand, he may be called 
Ebenezer Wilkes Smith ; on the other hand, he may not. But in 
a long life, during which I have met a great many strangers, I 
have only once found the former possibility realized. The pure 
mathematical theory, which merely enumerates possible cases, is 
devoid of practical interest unless we know that each possible 
case occurs approximately with equal frequency, or with some 
known frequency. And this, if we are considering events, not a 
logical schema, can only be known through actual statistics, the 
use of which, it may be said, must proceed more or less as in 
Reichenbach’s theory. 

This argument, also, I shall admit provisionally; it will be 
examined afresh when we come to consider induction. 

There is an objection of a quite different kind to Reichenbach’s 
theory as he states it, and that has to do with his introducing 
series where only classes seem to be logically relevant. Let us 
take an illustration : what is the chance that an integer chosen at 
random will be a prime ? If we take the integers in their natural 
order, the chance, on his definition, is zero; for if n is an integer, 
the number of primes less than or equal to n is approximately 
nflogn if n is large, so that the chance of an integer less than n 
being a prime approximates to 1 /logw, and the limit of 1 /log n as 
i» is indefinitely increased is zero. But now suppose we rearrange 
the integers on the following plan: Put first the first 9 primes, 
then the first number that is not a prime, then the next 9 primes, 
and then the second number that is not a prime, and so on 
indefinitely. When the integers are arranged in this order, 
Reichenbach’s definition shows that the chance of a number 
selected at random being a prime is We could even arrange 
the integers so that the chance of a number not being a prime 
would be zero. To get this result, begin with the first non-prime, 
i.e. 4, and put after the n th number which is not a prime the n 
primes next after those already placed; this series begins 4, 1,6, 

*, 3 * 8 > Sf 7» 9* I 3> l 7 > l 9 > 33 > *<>> 29, 31, 37, 41, 43, 12 



In this arrangement, there will be, before the (n + i) tt non- 
prime, n non-primes and J » (» + i) primes; thus as n increases, 
the ratio of the number of non-primes to the number of primes 
approaches o as a limit. 

From this illustration it is obvious that, if Reichenbach’s 
definition is accepted, given any class A having as many terms 
as there are natural numbers, and given any infinite sub-class B, 
the chance that an A selected at random will be a B will be 
anything from o to i (both included), according to the way in 
which we choose to distribute the B’s among the A’s. 

It follows that, if probability is to apply to infinite collections, 
it must apply to series, not to classes. This seems strange. 

It is true that, where empirical data are concerned, they are all 
given in a time-order, and therefore as a series. If we choose to 
assume that there is going to be an infinite number of events of 
the kind we are investigating, then we can also decide that our 
definition of probability is to apply only so long as the events 
are arranged in temporal sequence. But outside pure mathematics 
no series are known to be infinite, and most are, as far as we can 
judge, finite. What is the chance that a man of 60 will die of 
cancer? Surely we can estimate this without assuming that the 
number of men who, before time ends, will have died of cancer, 
is infinite. But according to the letter of Reichenbach’s definition 
this should be impossible. 

If probabilities depend upon taking events in their temporal 
order, rather than in any other order of which they are susceptible, 
then probability cannot be a branch of logic, but must be part 
of the study of the course of nature. This is not Reichenbach’s 
view; he holds, on the contrary, that all true logic is probability- 
logic, and that the classical logic is at fault because it classifies 
propositions as true or false, not as having this or that degree 
of probability. He should, therefore, be able to state what is 
fundamental in probability-theory in abstract logical terms, with- 
out introducing accidental features of the actual world, such as 

There is great difficulty in combining a statistical view of 
probability with the view, which Reichenbach also holds, that 
all propositions are only probable in varying degrees that fall 
short of certainty. The difficulty is that we seem committed to 
an endless regress. Suppose we say it is probable that a man 



HUMAN knowledge: its scope and limits 

We are not interested in the mere fact that there are 36 possible 
results of a throw with two dice. What we are interested in is the 
fact (if it be a fact) that in a long series of throws each of these 
36 possibilities will be realized an approximately equal number 
of times. This is a fact which does not follow from the mere 
existence of 36 possibilities. When you meet a stranger, there are 
exactly two possibilities: on the one hand, he may be called 
Ebenezer Wilkes Smith ; on the other hand, he may not. But in 
a long life, during which I have met a great many strangers, I 
have only once found the former possibility realized. The pure 
mathematical theory, which merely enumerates possible cases, is 
devoid of practical interest unless we know that each possible 
case occurs approximately with equal frequency, or with some 
known frequency. And this, if we are considering events, not a 
logical schema, can only be known through actual statistics, the 
use of which, it may be said, must proceed more or less as in 
Reichenbach’s theory. 

This argument, also, I shall admit provisionally; it will be 
examined afresh when we come to consider induction. 

There is an objection of a quite different kind to Reichenbach’s 
theory as he states it, and that has to do with his introducing 
series where only classes seem to be logically relevant. Let us 
take an illustration : what is the chance that an integer chosen at 
random will be a prime ? If we take the integers in their natural 
order, the chance, on his definition, is zero ; for if n is an integer, 
the number of primes less than or equal to n is approximately 
n/logn if n is large, so that the chance of an integer less than n 
being a prime approximates to i/logw, and the limit of i/log n as 
n is indefinitely increased is zero. But now suppose we rearrange 
the integers on the following plan: Put first the first 9 primes, 
then the first number that is not a prime, then the next 9 primes, 
and then the second number that is not a prime, and so on 
indefinitely. When the integers are arranged in this order, 
Reichenbach’s definition shows that the chance of a number 
selected at random being a prime is We could even arrange 
the integers so that the chance of a number not being a prime 
would be zero. To get this result, begin with the first non-prime, 
i.e. 4, and put after the n th number which is not a prime the n 
primes next after those already placed; this series begins 4, 1,6, 

*, 3» 8 » 5. 7. *«. 9, 13. 17. 19. 23. i°» 29, 31. 37. 4*. 43. ** 



In this arrangement, there will be, before the (n -f i) tt non- 
prime, n non-primes and J n (n + i) primes; thus as n increases, 
the ratio of the number of non- primes to the number of primes 
approaches o as a limit. 

From this illustration it is obvious that, if Reichenbach’s 
definition is accepted, given any class A having as many terms 
as there are natural numbers, and given any infinite sub-class B, 
the chance that an A selected at random will be a B will be 
anything from o to i (both included), according to the way in 
which we choose to distribute the B’s among the A’s. 

It follows that, if probability is to apply to infinite collections, 
it must apply to series, not to classes. This seems strange. 

It is true that, where empirical data are concerned, they are all 
given in a time-order, and therefore as a series. If we choose to 
assume that there is going to be an infinite number of events of 
the kind we are investigating, then we can also decide that our 
definition of probability is to apply only so long as the events 
are arranged in temporal sequence. But outside pure mathematics 
no series are known to be infinite, and most are, as far as we can 
judge, finite. What is the chance that a man of 60 will die of 
cancer? Surely we can estimate this without assuming that the 
number of men who, before time ends, will have died of cancer, 
is infinite. But according to the letter of Reichenbach’s definition 
this should be impossible. 

If probabilities depend upon taking events in their temporal 
order, rather than in any other order of which they are susceptible, 
then probability cannot be a branch of logic, but must be part 
of the study of the course of nature. This is not Reichenbach’s 
view; he holds, on the contrary, that all true logic is probability- 
logic, and that the classical logic is at fault because it classifies 
propositions as true or false, not as having this or that degree 
of probability. He should, therefore, be able to state what is 
fundamental in probability-theory in abstract logical terms, with- 
out introducing accidental features of the actual world, such as 

There is great difficulty in combining a statistical view of 
probability with the view, which Reichenbach also holds, that 
all propositions are only probable in varying degrees that fall 
short of certainty. The difficulty is that we seem committed to 
an endless regress. Suppose we say it is probable that a man 

385 BB 

human knowledge: its scope and limits 

who has plague will die of it. This means that, if we could 
ascertain the whole series of men who, from the earliest times 
till the extinction of the human race, will have suffered from 
plague, we should find that more than half of them will have died 
of it. Since the future and much of the past are unrecorded, we 
assume that the recorded cases are a fair sample. But now we 
are to remember that all our knowledge is only probable ; therefore 
if, in compiling our statistics, we find it recorded that Mr. A 
had plague and died of it, we must not regard this item as certain, 
but only as probable. To find out how probable it is, we must 
include it in a series, say of official death certificates, and we must 
find some means of ascertaining what proportion of death cer- 
tificates are correct. Here a single item in our statistics will be: 
“Mr. Brown was officially certified to have died, but turned out 
to be still alive”. But this, in turn, is to be only probable, and 
must therefore be one of a series of recorded official errors, some 
of which turned out to be not errors. That is to say, we must 
collect cases where it was falsely believed that a person certified 
dead had been found to be still alive. To this process there can 
be no end, if all our knowledge is only probable, and probability 
is only statistical. If we are to avoid an endless regress, and if 
all our knowledge is to be only probable, “probability” will have 
to be interpreted as “degree of credibility”, and will have to 
be estimated otherwise than by statistics. Statistical probability 
can only be estimated on a basis of certainty, actual or 

I shall return to Reichenbach in connection with induction. 
For the present, I wish to make clear my own view as to the 
connection of mathematical probability with the course of nature. 
Let us take as an illustration a case of Bernoulli's law of large 
numbers, choosing the simplest possible case. We have seen that, 
if we make up all possible integers consisting of n digits, each 
either i or 2, then, if n is large — say not less than 1,000 — a vast 
majority of the possible integers have an approximately equal 
number of i’s and 2’s. This is merely an application of the fact 
that, in the binomial expansion of (x + y) n , when n is large the 
sum of the coefficients near the middle falls not far short of the 
sum of all the coefficients, which is 2 n . But what has this to do 
with the statement that, if I toss a penny often, I shall probably 
get an approximately equal number of heads and tails ? The one 



is a logical fact, the other, apparently, an empirical fact; what is 
the connection between them ? 

With some interpretations of “probability”, a statement con- 
taining the word “probable” can never be an empirical statement. 
It is admitted that what is improbable may happen, and what 
is probable may fail to happen. It follows that what does happen 
does not show that a previous judgment of probability was either 
right or wrong; every imaginable course of events is logically 
compatible with every imaginable anterior estimate of probabili- 
ties. This can only be denied by maintaining that what is very 
improbable does not happen, which we have no right to maintain. 
In particular, if induction asserts only probabilities, then what- 
ever may happen is logically compatible both with the truth and 
with the falsehood of induction. Therefore the inductive principle 
has no empirical content. This is a reductio ad absurdum , and 
shows that we must connect the probable with the actual more 
closely than is sometimes done. 

If we adhere to the finite frequency theory — and so far I have 
seen no reason for not doing so — we shall say that, if we assert 
“a is an A” to be probably given “a is a B”, we mean that, in 
fact, most members of B are members of A. This is a statement 
of fact, but not a statement about a. And if I say that an inductive 
argument (suitably formulated and limited) makes its conclusion 
probable, I mean that it is one of a class of arguments, most of 
which have conclusions that are true. 

What, now, can I mean when I say that the chance of heads 
is a half? To begin with, this, if true, is an empirical fact; it does 
not follow from the fact that, in tossing a coin, there are only 
two possibilities, heads and tails. If it did, we could infer that 
the chance of a stranger being called Ebenezer Wilkes Smith is 
a half, since there are only two alternatives, that he is so called 
or that he isn’t. With some coins, heads come oftener than tails; 
with others, tails oftener than heads. When I say, without speci- 
fying the coin, that the chance of heads is a half, what do I mean ? 

My assertion, like all other empirical assertions that pretend 
to numerical exactitude, must be only approximate. When I say 
that a man’s height is 6 ft. i in., I am allowed a margin of 
error; even if I have said it on oath, I cannot be convicted of 
perjury if it turns out that I am a hundredth of an inch out. 
Similarly, I must not be held to have made a false statement 


HUMAN knowledge: its scope and limits 

about the penny if it turns out that 0*500001 would have been 
a more accurate estimate than 0*5. It is doubtful, however, 
whether any evidence could make me think 0*500001 a better 
estimate than 0*5. In probability, as elsewhere, we take the 
simplest hypothesis which approximately fits the facts. Take (say) 
the law of falling bodies. Galileo made a certain number of obser- 
vations, which fitted more or less with the formula s = \gt 2 . No 
doubt he could have found a function f(t) such that s = f(t) would 
have fitted his observations more exactly, but he preferred a 
simple formula which fitted well enough. 1 In the same way, if 
I tossed a coin 2,000 times and got 999 heads and 1,001 tails, 
I should take the chance of heads to be a half. But what exactly 
should I mean by this statement ? 

This question shows the strength of Reichenbach’s definition. 
According to him, I mean that, if I continue long enough, the 
proportion of heads will come, in time, to be permanently very 
near \ ; in fact, it will come to differ from £ by less than any 
fraction however small. This is a prophecy; if it is correct, my 
estimate of the probability is correct, but if not, not. What can 
the finite frequency theory oppose to this ? 

We must distinguish between what the probability is and what 
it probably is. As to what the probability is, that depends upon 
the class of tosses we are considering. If we are considering tosses 
with a given coin, then if, in the whole of its existence, this coin 
is going to have given m heads out of a total of n tosses, the 
probability of heads with that coin is m\n. If we are considering 
coins in general, n will have to be the total number of tosses of 
coins throughout the past and future history of the world, and 
m the number of these that will have been heads. We may, to 
make the problem less vast, confine ourselves to tosses this year 
in England, or to tosses tabulated by students of probability. In 
all these cases m and n are finite numbers, and min is the proba- 
bility of heads with the given conditions. 

But none of the above probabilities are known. We are there- 
fore driven to make estimates of them, that is to say, to find some 
way of deciding what they probably are. If we are to adhere to 
the finite frequency theory, this will mean that our series of heads 
and tails must be one of some finite class of series, and that we 
must have some relevant knowledge about this whole class. We 
1 Cf. Jeffreys, Theory of Probability , and Scientific Inference. 



will suppose it to have been observed that, in every series of 
10,000 or more tosses with a given coin, the proportion of heads 
after the 5,ooo /A toss has never varied by more than 2€, where € 
is small. We can then say : In every observed case, the proportion 
of heads after 5,000 tosses with a given coin has always remained 
between p — e and p + e, where p is a constant depending on 
the coin. To argue from this to a case not yet observed is a matter 
of induction. If this is to be valid, we shall need an axiom to the 
effect that (in certain circumstances) a characteristic which is 
present in all observed cases is present in a large proportion of 
all cases; or, at any rate, we shall need some axiom from which 
this results. We shall then be able to infer a probable probability 
from observed frequencies, interpreting probability in accordance 
with the finite frequency theory. 

The above is only an outline suggestion of a theory. The main 
point that I wish to emphasize is that, on the theory I advocate, 
every probability statement (as opposed to a merely doubtful 
statement) is a statement of fact , as to some proportion in a series. 
In particular, the inductive principle, whether true or false, will 
have to assert that, as a fact, most series of certain kinds have, 
throughout, any characteristic of a certain sort which is present 
in a large number of successive terms of the series. If this is a 
fact, inductive arguments may yield probabilities; if not, not. 
I do not at present inquire how we are to know whether it is 
a fact or not; that is a problem which I shall not consider until 
the last section of our inquiry. 

It will be seen that, in the above discussion, we have been led 
to agree with Reichenbach on many points, while consistently 
disagreeing as to the definition of probability. The main objection 
which I feel to his definition is that the frequency on which it 
depends is hypothetical and for ever unascertainable. I disagree 
also in distinguishing more sharply than he does between proba- 
bility and doubtfulness, and in holding that probability- logic is 
not logically the fundamental kind, as opposed to certainty-logic. 

3 g 9 

Chapter V 


K eynes’s Treatise on Probability (1921) sets out a theory 
which is, in a sense, the antithesis of the frequency theory. 
He holds that the relation used in deduction, namely 
“p implies q } \ is the extreme form of a relation which might be 
called “p more or less implies q”. “If a knowledge of h ”, he says, 
“justifies a rational belief in a of degree a, we say there is a 
probability relation of degree a between a and A”. We write 
this: “ ajh = a”. “Between two sets of propositions there exists 
a relation, in virtue of which, if we know the first, we can attach 
to the latter some degree of rational belief.” Probability is essen- 
tially a relation: “It is as useless to say 'b is probable’ as ‘A is 
equal’ or ‘A is greater than’.” From “a” and “ a implies A”, we 
can conclude “A”, that is to say, we can drop all mention of the 
premiss and simply assert the conclusion. But if a is so related 
to b that a knowledge of a renders a probable belief in b rational, 
we cannot conclude anything whatever about b which has not 
reference to a\ there is nothing corresponding to the dropping 
of a true premiss in demonstrative inference. 

Probability, according to Keynes, is a logical relation, which 
cannot be defined, unless, perhaps, in terms of degrees of rational 
belief. But on the whole it would seem that Keynes inclines rather 
to defining “degrees of rational belief” in terms of the probability- 
relation. Rational belief, he says, is derivative from knowledge: 
when we have a degree of rational belief in p, it is because we 
know some proposition h and also know p/A = a. It follows that 
some propositions of the form “p/A = a” must be among our 
premisses. Our knowledge is partly direct, partly by argument; 
our knowledge by argument proceeds through direct knowledge 
of propositions of the form “p implies q” or “ qjp = a”. In every 
argument, when fully analysed, we must have direct knowledge 
of the relation of the premisses to the conclusion, whether it be 
that of implication or that of probability in some degree. Know- 
ledge of A and of p/A = a leads to a “rational belief of the appro- 
priate degree” in p. Keynes explicitly assumes that all direct 
knowledge is certain, and that a rational belief which falls short 



of certainty can only arise through perception of a probability- 

Probabilities in general, according to Keynes, are not numeri- 
cally measurable; those that are so form a very special class of 
probabilities. He holds that one probability may not be com- 
parable with another, i.e. may be neither greater nor less than 
the other, nor yet equal to it. He even holds that it is sometimes 
impossible to compare the probabilities of p and not-/> on given 
evidencj. He does not mean that we do not know enough to do 
this; he means that there actually is no relation of equality or 
inequality. He thinks of probabilities according to the following 
geometrical scheme: Take two points, representing the o of 
impossibility and the i of certainty; then numerically measurable 
possibilities may be pictured as lying on the straight line between 

0 and i , while others lie on various curved routes from o to i . Of 
two probabilities on the same route, we can say that the one nearer 

1 is the greater, but we cannot compare probabilities on different 
routes, except when two routes intersect, which may happen. 

Keynes needs, as we have seen, some direct knowledge of 
probability-propositions. In order to make a beginning in obtain- 
ing such knowledge, he examines and emends what is called the 
“principle of non-sufficient reason”, or, as he prefers to call it, 
the “principle of indifference”. 

In its crude form, the principle states that if there is no known 
reason for one rather than another of several alternatives, then 
these alternatives are all equally probable. In this form, as he 
points out, the principle leads to contradictions. Suppose, for 
instance, you know nothing of the colour of a certain book ; then 
the chances of its being blue or not blue are equal, and therefore 
each is 1/2. Similarly the chance of its being black is 1/2. There- 
fore the chance of its being blue or black is 1. It follows that all 
books are either blue or black, which is absurd. Or suppose we 
know that a certain man inhabits either Great Britain or Ireland, 
shall we take these as our alternatives, or shall we take England, 
Scotland, and Ireland, or shall we take each county as equally 
probable? Or, if we know that the specific gravity of a certain 
substance lies between 1 and 3, shall we take the intervals 1 to 2 
and 2 to 3 as equally probable ? But if we consider specific volume, 
the intervals 1 to 2/3 and 2/3 to 1/3 would be the natural choice, 
which would make the specific gravity have equal chances of 


HUMAN knowledge: its scope and limits 

being between i and 3/2 or between 3/2 and 3. Such paradoxes 
can be multiplied indefinitely. 

Keynes does not, on this account, totally abandon the principle 
of indifference ; he thinks it can be so stated as to avoid the above 
difficulties and still be useful. For this purpose, he first defines 

Roughly speaking, an added premiss is “irrelevant” if it does 
not change the probability; i.e. A x is irrelevant in relation to 
x and A if xjhfi = x/h. Thus, for example, the fact that a man’s 
surname begins with M is irrelevant in estimating his chances 
of death. The above definition is, however, somewhat too simple, 
because h x might consist of two parts, of which one increased 
the probability of x while the other diminished it. For example: 
a white man’s chances of life are diminished by living in the 
tropics, but are increased (or so they say) by being a teetotaller. 
It may be that the death-rate among white teetotallers in the 
tropics is the same as that of white men in general, but we should 
not say that being a teetotaller who lives in the tropics was 
irrelevant. Therefore we say that h x is irrelevant to x/h if there 
is no part of h x which alters the probability of x. 

Keynes now 7 states the principle of indifference in the following 
form: The probabilities of a and b relative to given evidence are 
equal if there is no relevant evidence relating to a without cor- 
responding evidence relating to b ; that is to say, the probabilities 
of a and b relative to the evidence are equal, if the evidence 
is symmetrical with respect to a and b. 

There is, however, still a somewhat difficult proviso to be 
added. “We must exclude those cases, in which one of the alter- 
natives involved is itself a disjunction of sub-alternatives of the 
same form” When this condition is fulfilled, the alternatives are 
called indivisible relatively to the evidence. Keynes gives a formal 
definition of “divisible” as follows : An alternative <f){a) is divisible, 
relatively to evidence A, if, given A, is equivalent to il <f>(b) 

or $c)”, where <f>(b) and <f>(c) are incompatible, but each possi- 
ble, when A is true. It is essential, here, that <£(a), <£(A), <f>(c) are all 
values of the same propositional function. 

Keynes thus finally accepts as an axiom the principle that, on 
given evidence, <f>(a) and <f>(b) are equally probable, if (1) the 
evidence is symmetrical with respect to a and A, (2) relatively to 
the evidence, <f>(a) and <f>(b) are indivisible. 



To the above theory empiricists might raise a general objection. 
They might say that the direct knowledge of probability relations 
which it demands is obviously impossible. Deductive demonstrative 
logic — so this argument might run — is possible because it consists 
of tautologies, because it merely re- states our initial stock of 
propositions in other words. When it does more than this — when, 
for instance, it infers “Socrates is mortal” from “all men are 
mortal”, it depends upon experience for the meaning of the word 
“Socrates”. Nothing but tautologies can be known independently 
of experience, and Keynes does not contend that his probability- 
relations are tautologous. How, then, can they be known? For 
clearly they are not known by experience, in the sense in which 
judgments of perception are so known; and it is admitted that 
some of them are not inferred. They would constitute, therefore, 
if admitted, a kind of knowledge which empiricism holds to be 

I have much sympathy with this objection, but I do not think 
we can consider it decisive. We shall find, when we come to 
discuss the principles of scientific inference, that science is impos- 
sible unless we have some knowledge which we could not have 
if empiricism, in a strict form, were true. In any case, we should 
not assume dogmatically that empiricism is true, though we are 
justified in trying to find solutions of our problems which are 
compatible with it. The above objection, therefore, though it may 
cause a certain reluctance to accept Keynes's theory, should not 
make us reject it outright. 

There is a difficulty on a question which Keynes seems not 
to have adequately considered, namely: Does probability in rela- 
tion to premisses ever confer rational credibility on the proposition 
which is rendered probable, and, if so, under what circumstances? 
Keynes says that it is as nonsensical to say “ p is probable” as 
to say “p is equal” or “p is greater than”. There is, according 
to him, nothing analogous to the dropping of a true premiss in 
deductive inference. Nevertheless, he says that, if we know A, and 
we also know p/A = a, we are entitled to give to p “rational 
belief in the appropriate degree”. But when we do so we are no 
longer expressing a relation of p to A; we are using this relation 
to infer something about p. This something we may call “rational 
credibility”, and we can say; “p is rationally credible to the 
degree a”. But if this is to be a true statement about p, not 



involving mention of A, then A cannot be arbitrary. For suppose 
pjh = a and p/h' = a , are we, supposing A and A' both known, 
to give to p the degree a or the degree a of rational credibility? 
It is impossible that both answers should be correct in any given 
state of our knowledge. 

If it is true that “probability is the guide of life”, then there 
must be, in any given state of our knowledge, one probability 
which attaches to p more vitally than any other, and this proba- 
bility cannot be relative to arbitrary premisses. We must say that 
it is the probability which results when h is taken to be all our 
relevant knowledge. We can say: Given any body of propositions 
constituting some person’s certain knowledge, and calling the 
conjunction of this body of propositions A, there are a number 
of propositions, not members of this body, which have proba- 
bility-relations to it. If p is such a proposition, and pfh— a, 
then a is the degree of rational credibility belonging to p for that 
person. We must not say that, if h' is some true proposition, 
short of A, which the person in question knows, and if p/h f = a', 
then, for that person , p has the degree of credibility a'; it will 
only have this degree of credibility for a person whose relevant 
knowledge is summed up by A'. All this, however, no doubt 
Keynes would admit. The objection is, in fact, only to a certain 
looseness of statement, not to anything essential to the theory. 

A more vital objection is as to our means of knowing such 
propositions as pjh — a. I am not now arguing a priori that we 
cannot know them; I am merely inquiring how we can. It will 
be observed that if “probability” cannot be defined, there must 
be probability- propositions which cannot be proved, and which, 
therefore, if we are to accept them, must be among the premisses 
of our knowledge. This is a general feature of all logically articu- 
lated systems. Every such system starts, of necessity, with an 
initial apparatus of undefined terms and unproved propositions. 
It is obvious that an undefined term cannot appear in an inferred 
proposition unless it has occurred in at least one of the unproved 
propositions ; but a defined term need not occur in any unproved 
proposition. For example: so long as there were held to be 
undefined terms in arithmetic, there had to be also unproved 
axioms: Peano had three undefined terms and five axioms. But 
when numbers and addition are defined logically, arithmetic needs 
no unproved propositions beyond those of logic. So, in our case, 


Keynes’s theory of probability 

if “probability” can be defined, it may be that all propositions 
in which the word occurs can be inferred; but if it cannot be 
defined, there must, if we are to know anything about it, be 
propositions, containing the word, which we know without 
extraneous evidence. 

It is not quite clear what sort of propositions Keynes would 
admit as premisses in our knowledge of probability. Do we 
directly know propositions of the form “p/A — a”? And when 
a probability is not numerically measurable, what sort of thing 
is a? Or do we only know equalities and inequalities, i.e. 
pjh <qjh t or p/A = qjh ? I incline to think that the latter is 
Keynes’s view. If so, the fundamental facts in the subject are 
relations of three propositions, not of two : we ought to start from 
a triadic relation 

P (p, q, h) 

meaning: given A, p is less probable than q. We might then say: 

“p/A = q/h” means : “Neither P(p, q , h) nor P(q f p, A)”. 

We should assume that P is asymmetrical and transitive with 
respect to p and q while h is kept constant. Keynes’s principle 
of indifference, if accepted, will then enable us, in certain cir- 
cumstances, to prove pjh = qjh. And from this basis the calculus 
of probabilities, in so far as Keynes considers it valid, can be 
built up. 

The above definition of equality can only be adopted if p\h 
and qjh are comparable; if (as Keynes holds possible) neither is 
greater than the other and yet they are not equal, the definition 
must be abandoned. We could meet this difficulty by axioms as 
to the circumstances under which two probabilities must be 
comparable. When they are comparable they lie on one route 
between o and i. On the right-hand side of the above definition 
of “p/A = qjh” we must then add that pjh and qlh are “com- 

Let us now re-state Keynes’s principle of indifference. He is 
concerned to establish circumstances in which pjh = qjh. This 
will happen, he says, if two conditions (sufficient but not neces- 
sary) are fulfilled. Let p be of the form <f>(a) and q of the 
form <f>(b) ; then h must be symmetrical with respect to a and b, 
and <f>(a), <f>{b) must be “indivisible”. 


HUMAN knowledge: its scope and limits 

When we say that h is symmetrical with respect to a and 6, 
we mean, presumably, that, if h is of the form f(a , b ), then 

/(*, b) — a). 

This will happen, in particular, if f(a , b) is of the form g(a). g(b) y 
which is the case when the information that h gives about a and b 
consists of separate propositions, one about a and the other about 6, 
and both are values of one propositional function. 

We now put p = <f>(a) } q = <f>(b ), h — f(a, b). 

Our axiom must be to the effect that, with a suitable proviso, 
the interchange of 0(a) and </>(b) cannot make any difference. This 
involves that b) = *(*)//(«, b) 

provided <f>(a) and <f>(b ) are comparable with respect to /(a , b). 
This follows if, as a general principle, 

0a/0 a = 0£/0 b 

that is to say, if probability depends not on the particular subject 
but on propositional functions. There seems hope, along these 
lines, of arriving at a form of the principle of indifference which 
might have more self-evidence than Keynes’s. 

Let us, for this purpose, examine his condition of indivisibility. 
Keynes defines “0(a) is divisible” as meaning that there are 
two arguments b and c such that “0a” is equivalent to “ <f>b or 
and <f>b and (j>c cannot both be true, while <f>b y (f>c are both 
possible given h. I do not think this is quite what he really wishes 
to say. We get nearer to what he wishes, I think, if we assume 
a and b and c to be classes, of which a is the sum of b and c. 
In that case, </> must be a function which takes classes as argu- 
ments. E.g. let a be an area on a target, divided into two parts 
b and c. Let “ <f>a ” be “some point of a is hit” and “0a” be 
“some point of a is aimed at”. Then (f>a is divisible in the above 
sense, and we do not have 

fa/tpa — <j>b jifjb 

for obviously <j>a jifja is greater than <f>b/ifjb. 

But it is not clear that our earlier condition, namely that h 
should be symmetrical with respect to a and b y does not suffice. 
For now h contains the proposition “6 is part of a”, which is 
not symmetrical. 



Keynes discusses the conditions for fa/ipa = <f>b/tfjb t and gives 
as an example of failure the case where <]>x . = . x is Socrates. 
In that case, no matter what ipx may be, 

^(Socrates) /^(Socrates) = i, 

while if b is not Socrates, </>ft/t fjb — o. 

To exclude this case, I should make the proviso that 
must not contain “a”. To take an analogous case, put 

</>x . = . x kills a, tfjx . = . x inhabits England. 

Then fajita is the likelihood of a committing suicide if English, 
whereas <f>xlipx, in general, is the likelihood of a being murdered 
by some Englishman who is named x. Obviously, in most cases, 
(jyaltjja is greater than <f>bjif)b y because a man is more likely to kill 
himself than to kill another person selected at random. 

The essential condition, then, seems to be that “<£#” must not 
contain “a” or “ft”. If this condition is fulfilled, I do not see 
how we can fail to have 

(f>a/ipa = <f>b/i/jb. 

I conclude that what the principle of indifference really asserts 
is that probability is a relation between propositional functions, 
not between propositions. This is what is meant by such phrases 
as “a random selection”. This phrase means that we are to con- 
sider a term solely as one satisfying a certain propositional 
function; what is said is, then, really about the propositional 
function and not about this or that value of it. 

Nevertheless, there remains something substantial which is 
what really concerns us. Given a probability- relation between two 
propositional functions (f>x and tpx, we can regard this as a 
relation between <f>a and ifja y provided “<£#” and do not 

contain “a”. This is a necessary axiom in all applications of 
probability in practice, for then it is particular cases that concern 

My conclusion is that the chief formal defect in Keynes's theory 
of probability consists in his regarding probability as a relation 
between propositions rather than between propositional functions. 
The application to propositions, I should say, belongs to the uses 
of the theory, not to the theory itself. 


Chapter VI 



T hat all human knowledge is in a greater or less degree 
doubtful is a doctrine that comes to us from antiquity; it 
was proclaimed by the sceptics, and by the Academy in its 
sceptical period. In the modern world it has been strengthened 
by the progress of science. Shakespeare, to represent the most 
ridiculous extremes of scepticism, says: 

Doubt that the stars are fire, 

Doubt that the sun doth move. 

The latter, when he wrote, had already been questioned by 
Copernicus, and was about to be even more forcibly questioned 
by Kepler and Galileo. The former is false, if “fire” is used in 
its chemical sense. Many things which had seemed indubitable 
have turned out to be in all likelihood untrue. Scientific theories 
themselves change from time to time, as new T evidence accumu- 
lates; no prudent man of science feels the same confidence in 
a recent scientific theory as was felt in the Ptolemaic theory 
throughout the middle ages. 

But although every part of what we should like to consider 
“knowledge” may be in some degree doubtful, it is clear that 
some things are almost certain, while others are matters of 
hazardous conjecture. For a rational man, there is a scale of 
doubtfulness, from simple logical and arithmetical propositions, 
and perceptive judgments, at one end, to such questions as what 
language the Myceneans spoke or “what song the Sirens sang”, 
at the other. Whether any degree of doubtfulness attaches to the 
least dubitable of our beliefs, is a question with which we need 
not at present concern ourselves ; it is enough that any proposition 
concerning which we have rational grounds for some degree of 
belief or disbelief can, in theory, be placed in a scale between 
certain truth and certain falsehood. Whether these limits are 
themselves to be included, we may leave an open question. 

There is a certain connection between mathematical probability 



and degrees of credibility. The connection is this: When, in 
relation to all the available evidence, a proposition has a certain 
mathematical probability, then this measures its degree of credi- 
bility. For instance, if you are about to throw dice, the proposition 
“double sixes will be thrown” has only one thirty-fifth of the 
credibility attaching to the proposition “double sixes will not be 
thrown”. Thus the rational man, who attaches to each proposition 
the right degree of credibility, will be guided by the mathematical 
theory of probability when it is applicable . 

The concept “degree of credibility”, however, is applicable 
much more widely than that of mathematical probability; I hold 
that it applies to every proposition except such as neither are 
data nor are related to data in any way which is favourable or 
unfavourable to their acceptance. I hold, in particular, that it 
applies to propositions that come as near as is possible to merely 
expressing data. If this view is to be logically tenable, we must hold 
that the degree of credibility attaching to a proposition is itself 
sometimes a datum. I think we should also hold that the degree 
of credibility to be attached to a datum is sometimes a datum, 
and sometimes (perhaps always) falls short of certainty. We may 
hold, in such a case, that there is only one datum, namely, a 
proposition with a degree of credibility attached to it, or we may 
hold that the datum and its degree of credibility are two 
separate data. I shall not consider which of these two views 
should be adopted. 

A proposition which is not a datum may derive credibility from 
various different sources ; a man who wishes to prove his innocence 
of a crime may argue both from an alibi and from his previous 
good character. The grounds in favour of a scientific hypothesis 
are practically always composite. If it is admitted that a datum 
may not be certain, its degree of credibility may be increased 
by an argument, or, on the contrary, may be rendered very small 
by a counter-argument. 

The degree of credibility conferred by an argument is not 
capable of being estimated simply. Take, first, the simplest 
possible case, namely that in which the premisses are certain and 
the argument, if valid, is demonstrative. At each step we have to 
“see” that the conclusion of this step follows from its premisses. 
Sometimes this is easy, for example if the argument is a syllogism 
in Barbara. In such a case, the degree of credibility attaching to 


HUMAN knowledge: its scope and limits 

the connection of premisses and conclusion is almost certainty, 
and the conclusion has almost the same degree of credibility as 
the premisses. But in the case of a difficult mathematical argument 
the chance of an error in reasoning is much greater. The logical 
connection may be completely obvious to a good mathematician, 
while to a pupil it is barely perceptible, and that only at moments. 
The pupil’s grounds for believing in the validity of the step are 
not purely logical; they are in part arguments from authority. 
These arguments are by no means demonstrative, for even the 
best mathematicians sometimes make mistakes. On such grounds, 
as Hume points out, the conclusion of a long argument has less 
certainty than the conclusion of a short one, for at each step there 
is some risk of error. 

By means of certain simplifying hypotheses, this source of 
uncertainty could be brought within the scope of the mathematical 
theory of probability. Suppose it established that, in a certain 
branch of mathematics, good mathematicians are right in a step 
in their arguments in a proportion x of all cases ; then the chance 
that they are right throughout an argument of n steps is It 
follows that a long argument which has not been verified by 
repetition runs an appreciable risk of error, even if x is nearly i. 
But repetition can reduce the risk until it becomes very small. All 
this is within the scope of the mathematical theory. 

What, however, is not within the scope of that theory is the 
private conviction of the individual mathematician as he takes 
each step. This conviction will vary in degree according to the 
difficulty and complexity of the step ; but in spite of this variability 
it must be as direct and immediate as our confidence in objects 
of perception. To prove that a certain premiss implies a certain 
conclusion, we must “see” each step ; we cannot prove the validity 
of the step except by breaking it up into smaller steps, each of 
which will then have to be “seen”. Unless this is admitted, all 
arguments will be lost in an endless regress. 

I have been speaking, so far, of demonstrative inference, but 
as regards our present question non-demonstrative inference 
presents no new problem, for, as we have seen, even demonstrative 
inference, when carried out by human beings, only confers 
probability on the conclusion. It cannot even be said that reasoning 
which professes to be demonstrative always confers a higher 
degree of probability on the conclusion than reasoning which is 


avowedly only probable; of this there are many examples in 
traditional metaphysics. 

If — as I believe, and as I shall argue in due course — data, as 
well as results of inference, may be destitute of the highest 
attainable degree of credibility, the epistemological relation 
between data and inferred propositions becomes somewhat com- 
plex. I may, for instance, think that I recollect something, but 
find reason to believe that what I seemed to recollect never 
happened; in that case I may be led by argument to reject a 
datum. Conversely, when a datum has, per se, no very high degree 
of credibility, it may be confirmed by extraneous evidence; for 
example, I may have a faint memory of dining with Mr. So-and-So 
some time last year, and may find that my diary for last year has 
an entry which corroborates my recollection. It follows that every 
one of my beliefs may be strengthened or weakened by being 
brought into relation with other beliefs. 

The relation between data and inferences, however, remains 
important, since the reason for believing no matter what must 
be found, after sufficient analysis, in data, and in data alone. (I 
am here including among data the principles used in any in- 
ferences that may be involved.) What does result is that the data 
relevant to some particular belief may be much more numerous 
than they appear to be at first sight. Take again the case of 
memory. The fact that I remember an occurrence is evidence, 
though not conclusive evidence, that the occurrence took place. 
If I find a contemporary record of the occurrence, that is con- 
firmatory evidence. If I find many such records, the confirmatory 
evidence is strengthened. If the occurrence is one which, like a 
transit of Venus, is made almost certain by a well-established 
scientific theory, this fact must be added to the records as an 
additional ground for confidence. Thus while there are beliefs 
which are only conclusions of arguments, there are none which, 
in a rational articulation of knowledge, are only premisses. In 
saying this, I am speaking in terms of epistemology, not of logic. 

Thus an epistemological premiss may be defined as a proposition 
which has some degree of rational credibility on its own account, 
independently of its relations to other propositions. Every such 
proposition can be used to confer some degree of credibility on 
propositions which either follow from it or stand in a probability 
relation to it. But at each stage there is some diminution of the 

401 CC 

human knowledge: its scope and limits 

original stock of credibility; the case is analogous to that of a 
fortune which is lessened by death duties on each occasion when 
it is inherited. Carrying the analogy a little further, we may say 
that intrinsic credibility is like a fortune acquired by a man’s 
own efforts, while credibility as the result of an argument is like 
inheritance. The analogy holds in that a man who has made a 
fortune can also inherit one, though every fortune must owe its 
origin to something other than inheritance. 

In this chapter I propose to discuss credibility, first in relation 
to mathematical probability, then in relation to data, then in 
relation to subjective certainty, and finally in relation to rational 


I am now concerned to discuss the question: In what circum- 
stances is the credibility of a proposition a derived from the 
frequency of ifix given some <j>x ? In other words, if “<£a” is 
“a is an a”, in what circumstances is the credibility of 
“ a is a /8” derived from one or more propositions of the form: 
“A proportion m/n of the members of a are members of /3”? This 
question, we shall find, is not quite so general as the one we 
ought to ask, but it will be desirable to discuss it first. 

It seems clear to common sense that, in the typical cases of 
mathematical probability, it is equal to degree of credibility. If 
I draw a card at random from a pack, the degree of credibility 
of “the card will be red” is exactly equal to that of “the card will 
not be red”, and therefore the degree of credibility of either is 1/2, 
if 1 represents certainty. In the case of a die, the degree of credi- 
bility of “1 will come uppermost” is exactly the same as that of 
“2 will come uppermost”, or 3 or 4 or 5 or 6. Hence all the 
derived frequencies of the mathematical theory can be interpreted 
as derived degrees of credibility. 

In this translation of mathematical probabilities into degrees 
of credibility, we make use of a principle which the mathematical 
theory does not need. The mathematical theory merely counts 
cases; but in the translation we have to know, or assume, that 
each case is equally credible. The need of this principle has long 
been recognized ; it has been called the principle of non-sufficient 
reason, or (by Keynes) the principle of indifference. We COn- 

'tfi . 



sidered this principle in connection with Keynes, but we must 
now consider it on its own account. Before discussing it, I wish 
to point out that it is not needed in the mathematical theory of 
probability. In that theory, we only need to know the numbers 
of various classes. It is only when mathematical probability is 
taken as a measure of credibility that the principle is required. 

What we need is something like the following: “Given an 
object a , concerning which we wish to know what degree of 
credibility to attach to the proposition ‘ a is a and given that 
the only relevant knowledge we have is { a is an a, then the 
degree of credibility of ‘ a is a 0’ is the mathematical probability 
measured by the ratio of the number of members common to a 
and P to the number of members of a”. 

Let us illustrate this by considering once more the tallest 
person in the United States, and the chance that he lives in Iowa. 
We have here, on the one hand, a description d y known to be 
applicable to one and only one of a number of named persons 
A x , A 2 , . . . A„, where n is the number of inhabitants of the United 
States. That is to say, one and only one of the propositions 
“ d = A,” (where r runs from i to n) is known to be true, but 
we do not know which. If this is really all our relevant knowledge, 
we assume that any one of the propositions “ d = A r ” is as 
credible as any other. In that case, each has a credibility i jn. 
If there are m inhabitants of Iowa, the proposition “ d inhabits 
Iowa” is equivalent to a disjunction of m of the propositions 
“ d = A/’, and therefore has m times the credibility of any one 
of them, since they are mutually exclusive. Therefore it has a 
degree of credibility measured by mjn. 

Of course in the above illustration the propositions “ d = A” 
are not all on a level. The evidence enables us to exclude children 
and dwarfs, and probably women. This shows that the principle 
may be difficult to apply, but does not show that it is false. 

The case of drawing a card from a pack comes nearer to 
realizing the conditions required by the principle. Here the 
description “d” is “the card I am about to draw”. The 52 cards 
all have what we may regard as names: “2 of spades”, etc. We 
have thus 52 propositions “</ — A r ”, of which one and only one 
is true, but we have no evidence whatever inclining us to one 
rather than another. Therefore the credibility of each is 1 /s 2. This, 
if admitted, connects credibility with mathematical probability. 



We may therefore enunciate, as a possible form of the “principle 
of indifference”, the following axiom : 

“Given a description d , concerning which we know that it is 
applicable to one and only one of the objects a lt a 2i . . . a nt and 
given that we have no knowledge bearing on the question which 
of these objects the description applies to, then the n propositions 
*d= a r * ( i < r < n) are all equally credible, and therefore each 
has a credibility measured by i /«.” 

This axiom is more restricted than the principle of non- 
sufficient reason as usually enunciated. We have to inquire 
whether it will suffice, and also whether we have reason to 
believe it. 

Let us first compare the above with Keynes’s principle of 
indifference, discussed in an earlier chapter. It will be re- 
membered that his principle says: the probabilities of p and q 
relative to given evidence are equal if (i) the evidence is 
symmetrical with respect to p and q , (2) p and q are “indi- 
visible”, i.e. neither is a disjunction of propositions of the 
same form as itself. We decided that this could be simplified: 
what is needed, we said, is that p and q should be values of one 
propositional function — say p— (j>(a) and q = <f>(b ) ; that 
should not contain either a or b\ and that, if the evidence contains 
a mention of a , say in the form ip(a) y it must also contain i/j(b) y 
and vice versa, where «/oc, in turn, must not mention a or b. This 
principle is somewhat more general than the one enunciated in 
the previous paragraph : it implies the latter, but I doubt whether 
the latter implies it. We may perhaps accept the more general 
principle, and re-state it as follows: 

“Given two propositional functions <f>x, ifjx , neither of which 
mentions a or 6, or, if it does so, mentions them symmetrically, 
then, given ipa and tpb, the two propositions 4 >a, <f>b have equal 

This principle, if accepted, enables us to infer credibility from 
mathematical probability, and makes all the propositions of the 
mathematical theory available for measuring degrees of credibility 
in the cases to which the mathematical theory is applicable. 

Let us apply the above principle to the case of n balls in a 
bag, each of which is known to be either white or black; the 
question is: what is the probability that there are x white balls? 
Laplace assumed that every value of x from o to n is equally 



likely, so that the probability of a given x is i /(« + i). From 
a purely mathematical standpoint, this is legitimate, provided 
we start from the propositional function : 

x = the number of white balls. 

But if we start from the propositional function : 

x is a white ball, 

we obtain a quite different result. In this case, there are many 
ways of choosing x balls. The first ball can be chosen in n ways; 
when it has been chosen, the next can be chosen in n — i ways, 
and so on. Thus the number of ways of choosing x balls is 

n times (w — i) times ( n — 2) times . . . times (n — x + 1). 
This is the number of ways in which there can be x white balls. 
To get the probability of x white balls, we have to divide this 
number by the sum of the numbers of ways of choosing o white 
balls, or 1, or 2, or 3, or . . . or n. This sum is easily shown to 
be 2 W . Therefore the chance of exactly x white balls is obtained 
by dividing the above number by 2 W . Let us call it “/>(w, r)”. 

This has a maximum when x = \n if n is even, or when 
x = \n ± \ if n is odd. Its value when x or n — x is small is 
very small if n is large. From the purely mathematical point of 
view, these two very different results are equally legitimate. But 
when we come to the measurement of degrees of credibility, there 
is a great difference between them. Let us have some way, inde- 
pendent of colour, by which we can distinguish the balls ; e.g. let 
them be successively drawn out of the bag, and let us call the 
one first drawn d u the one drawn second </ 2 > and so on. Put “0” 
for “white” and “A” for “black”, and put “<£0” for “white is 
the colour of d^\ “<£&” for “black is the colour of d The 
evidence is that cf>a or <f>b is true, but not both. This is symmetrical, 
and therefore, on the evidence, <j>a and <f>b have equal credibility, 
i.e. “dj is white” and “d { is black” have equal credibility. The 
same reasoning applies to d 2y rf 3 , . . . d n . Thus in the case of each 
ball the degrees of credibility of white and black are equal. And 
therefore, as a simple calculation shows, the degree of credibility 
of * white balls is p(n , x ), where it is assumed that x lies between 
o and w, both included. 

It is to be observed that, in measuring degrees of credibility, 
we suppose the data not only true, but exhaustive in relation to 


human knowledge: its scope and limits 

our knowledge, i.e. we assume that we know nothing relevant 
except what is mentioned in the data. Therefore for a given person 
at a given time there is only one right value for the degree of 
credibility of a given proposition, whereas in the mathematical 
theory many values are equally legitimate in relation to many 
different data, which may be purely hypothetical. 

In applying the results of the mathematical calculus of proba- 
bility to degrees of credibility, we must be careful to fulfil two 
conditions. First, the cases which form the basis of the mathe- 
matical enumeration must all be equally credible on the evidence; 
second, the evidence must include all our relevant knowledge. 
As to the former of these conditions a few words must be said. 

Every mathematical calculation of probability starts from some 
fundamental class, such as a certain number of tosses of a coin, 
a certain number of throws of a die, a pack of cards, a collection 
of balls in a bag. Each member of this fundamental class counts 
as one. From it we manufacture other logically derivative classes, 
e.g. a class of n series of ioo tosses of a coin. Out of these n series 
we can pick out the sub-class of those that consist of 50 heads 
and 50 tails. Or, starting from a pack of cards, we can consider 
the class of possible “hands”, i.e. selections of 13 cards, and 
proceed to inquire how many of these contain 11 cards of one 
suit. The point is that the frequencies that are calculated always 
apply to classes having some structure logically defined in relation 
to the fundamental class, whereas the fundamental class, for the 
purposes of the problem, is regarded as composed of members 
having no logical structure, i.e. their logical structure is irrelevant. 

So long as we confine ourselves to the calculation of frequencies, 
i.e. to the mathematical theory of probability, we can take any 
class as our fundamental class, and calculate frequencies in relation 
to it. It is not necessary to make an assumption to the effect that all 
the members of the class are equally probable; all that we need 
to say is that, for the purpose in hand, each member of the class 
is to count as one. But when we wish to ascertain degrees of 
credibility, it is necessary that our basic class should consist of 
propositions which are all equally credible in relation to the 
evidence. Keynes’s “indivisibility” is intended to secure this. I 
should prefer to say that the members of the fundamental class 
must have “relative simplicity”, i.e. they must not have a structure 
definable in terms of the data. Take, e.g., white and black balls 



in a bag. Each ball has, in fact, an incredibly complicated struc- 
ture, since it consists of billions of molecules; but this is quite 
irrelevant to our problem. On the other hand, a collection of m 
balls chosen from a fundamental class of n balls has a logical 
structure relatively to the fundamental class. If each member of 
the fundamental class has a name, every sub-class of m terms 
can be defined. All calculations of probability have to do with 
classes which can be defined in terms of the fundamental class. 
But the fundamental class itself must consist of members which 
cannot be logically defined in terms of the data. I think that when 
this condition is fulfilled the principle of indifference is always 

At this point, however, a caution is necessary. There are two 
ways in which “a is an a” may become probable, either (i) because 
it is certain that a belongs to a class most of which are a’s, or 
(2) because it is probable that a belongs to a class all of which 
are a’s. For instance, we may say “Mr. A. is probably mortal” 
if we are sure that most men are mortal, or if we have reason 
to think it probable that all men are mortal. When we make a 
throw with two dice, we can say “probably we shall not throw 
double sixes”, because we know that most throws are not double 
sixes. On the other hand, suppose I have evidence suggesting, 
but not proving, that a certain bacillus is always present in a 
certain disease; I may then say, in a given case of this disease, 
that probably the bacillus in question is present. There is in each 
case a kind of syllogism. In the first case, 

Most A is B ; 

This is an A; 

Therefore this is probably a B. 

In the second case, 

Probably all A is B ; 

This is an A ; 

Therefore this is probably a B. 

The second case, however, is more difficult to reduce to a fre- 
quency. Lgt us inquire whether this is possible. 

In some cases, this is clearly possible. E.g. most \Vords do not 
contain the letter Z. Therefore, if some word is chosen at hap- 
hazard, it is probable that all its letters are other than Z. Thus if 


HUMAN knowledge: its scope and limits 

A = the class of letters in the word in question, and B = the 
class of letters other than Z, we get a case of our second pseudo- 
syllogism. The word, of course, must be defined in some way 
which leaves us in temporary ignorance as to what it is, e.g. the 
8,ooo** word in Hamlet , or the third word on p. 248 of the 
Concise Oxford Dictionary . Assuming that you do not at present 
know what these words are, you will be wise to bet against their 
containing a Z. 

In all cases of our second pseudo-syllogism, it is clear that 
what I have been calling the “fundamental class” is given as 
a class of classes, and therefore its logical structure is essential. 
To generalize the above instance : let k be a class of classes, such 
that most of its members are entirely contained in a certain 
class / 3 ; then, from *‘x is an a” and “a is a we can conclude 
“ x is probably a £”. (In the above instance, k was the class of 
words, a the class of letters in a certain word, and the alphabet 
without Z.) The odd thing is that, denoting by “sum of #c” the 
class of members of members of k, our premisses do not suffice 
to prove that a member of the sum of k is probably a member 
of j8. For example, let k consist of the three words STRENGTH, 
QUAIL, MUCK, together with all words containing no letter 
occurring in any of these three. Then the sum of k consists of 
all the letters of the alphabet, possibly excepting Z. 1 But “ x is 
an a and a is a /c” makes it probable that x is not one of the 
letters occurring in the above three words, while “ x is a member 
of the sum of k” does not make this probable. This illustrates 
the complications that arise when the fundamental class has a 
structure which is relevant to the probabilities. But in such cases 
as the above it is still possible to measure credibility by frequency, 
though less simply. 

There is, however, another and more important class of cases, 
which we cannot adequately discuss except in connection with 
induction. These are the cases where we have inductive evidence 
making it probable that all A is B, and we infer that a particular 
A is probably a B, e.g. probably all men are mortal ( not all men 
are probably mortal), therefore Socrates is probably mortal. This 
is a pseudo-syllogism of our second kind. But if the “probably” 
in “probably all men are mortal” can be reduced to a frequency, 

1 Whether Z is to be included depends upon whether “ZOO” is 
allowed to count as a word. 



it certainly cannot be so reduced at all simply. I will therefore 
leave this class of cases to be discussed at a later stage. 

There are, we shall find, various examples of degrees of credi- 
bility not derivable from frequencies. These I shall now proceed 
to consider. 


In the present section I propose to advocate an unorthodox 
opinion, namely, that a datum may be uncertain. There have been 
hitherto two views: first, that in a proper articulation of know- 
ledge we start from premisses which are certain in their own 
right, and may be defined as “data”; second, that, since no know- 
ledge is certain, there are no data, but our rational beliefs form 
a closed system in which each part lends support to every other 
part. The former is the traditional view, inherited from the 

Greeks, enshrined in Euclid and theology; the latter is a view 

first advocated, if I am not mistaken, by Hegel, but most 

influentially supported, in our day, by John Dewey. The view 

which I am about to set forth is a compromise, but one somewhat 
more in favour of the traditional theory than of that advocated 
by Hegel and Dewey. 

I define a “datum” as a proposition which has some degree 
of rational credibility on its own account, independently of any 
argument derived from other propositions. It is obvious that the 
conclusion of an argument cannot derive from the argument a 
higher degree of credibility than that belonging to the premisses ; 
consequently, if there is such a thing as rational belief, there 
must be rational beliefs not wholly based on argument. It does 
not follow that there are beliefs which owe none of their credibility 
to argument, for a proposition may be both inherently credible 
and also a conclusion from other propositions that are inherently 
credible. But it does follow that every proposition which is 
rationally credible in any degree must be so either (a) solely in 
its own right, or (6) solely as the conclusion from premisses which 
are rationally credible in their own right, or (c) because it has 
some degree of credibility in its own right, and also follows, by 
a demonstrative or probable inference, from premisses which 
have some degree of credibility in their own right. If all proposi- 
tions which have any credibility in their own right are certain, 
case (c) has no importance, since no argument can make such 


human knowledge: its scope and limits 

propositions more certain. But on the view which I advocate, 
case (c) is of the greatest importance. 

The traditional view is adopted by Keynes, and set forth by 
him in his Treatise on Probability , p. 16. He says: 

“In order that we may have a rational belief in p of a lower 
degree of probability than certainty, it is necessary that we know 
a set of propositions A, and also know some secondary proposition 
q asserting a probability- relation between p and A. 

“In the above account one possibility has been ruled out. It 
is assumed that we cannot have a rational belief in p of a degree 
less than certainty except through knowing a secondary proposi- 
tion of the prescribed type. Such belief can only arise, that is 
to say, by means of the perception of some probability- relation. 
. . . All knowledge which is obtained in a manner strictly direct 
by contemplation of the objects of acquaintance and without any 
admixture whatever of argument and the contemplation of the 
logical bearing of any other knowledge on this, corresponds to 
certain rational belief and not to a merely probable degree of 
rational belief.” 

I propose to controvert this view. For this purpose I shall 
consider (1) faint perception, (2) uncertain memory, (3) dim 
awareness of logical connection. 

(1) Faint perception . — Consider such familiar experiences as 
the following, (a) You hear an aeroplane going away; at first 
you are sure you hear it, and at last you are sure you do not 
hear it, but in the interval there is a period during which you are 
not sure whether you still hear it or not. ( b ) You are watching 
Venus during the dawn ; at first you see the planet shining brightly, 
and at last you know that daylight has made it invisible, but 
between these two times you may be in doubt whether you are 
still seeing it or not. (c) In the course of travel you have attracted 
a number of fleas; you set to work to get rid of them, and in 
the end you are sure you have succeeded, but in the meantime 
you are troubled by occasional doubtful itches. ( d ) By mistake 
you make tea in a pot that has contained vinegar; the result is 
appalling. You rinse the pot and try again, but still the offensive 
flavour is unmistakeable. After a second rinsing you are doubtful 
whether you still taste the vinegar; after a third you are sure 
you do not. (e) Your drains are out of order, and you call in 
the plumber. At first, after his visit, you feel sure that the offensive 



odour is gone, but gradually, through varying stages of doubt, you 
become certain that it has returned. 

Such experiences are familiar to every one, and must be taken 
account of in any theory as to the knowledge based on sense- 

(2) Uncertain memory .— In The Tempest (Act I, Scene II), 
Prospero asks Miranda to look into “the dark backward and 
abysm of time”; she says “had I not four or five women once 
that tended me?” and Prospero confirms her doubtful recollec- 
tion. We all have memories of this kind, about which we do not 
feel sure. Usually, if it is worth while, we can discover from other 
evidence whether they are veridical or not, but that is irrelevant 
to our present thesis, which is that they have a certain degree 
of credibility on their own account, though this degree may fall 
far short of full certainty. A recollection which has a fairly high 
degree of credibility contributes its quota to our grounds for 
believing in some past occurrence for which we have other 
evidence. But here a distinction is necessary. The past event I 
uncertainly remembered has partial credibility in itself ; but when 
I adduce the recollection as a ground for belief, I am no longer 
treating the past occurrence as a datum, for it is not it but the 
present recollecting that is my datum. My recollecting confers 
some credibility on what is recollected; how much credibility, 
we can more or less ascertain inductively by a statistical inquiry 
into the frequency of errors of memory. But this is a different 
matter from past occurrences as data. That such data must be 
supplied by memory is a thesis which I have argued elsewhere. 

(3) Dim awareness of logical connection . — Any person whose 
mathematical abilities are not almost superhuman must, if he has 
studied mathematics, have often had the experience of being 
hardly able to “see” a certain step in a proof. The process of 
following a proof is facilitated by making the steps very small, 
but however small we make them some of them may remain 
difficult if the subject-matter is very complex. It is obvious that, 
if we have made the steps as small as possible, each step must 
be a datum, for otherwise every attempt at proof would involve 
an endless regress. Consider, say, a syllogism in Barbara. I say 
“all men are mortal”, and you agree. I say “Socrates is a man”, 
and you agree. I then say “therefore Socrates is mortal”, and 
you say “I don't see how that follows”. What, then, can I do? 

41 1 

HUMAN knowledge: its scope and limits 

I can say: “Don’t you see that if f(x) is always true, then f(a) 
is true? and don’t you see that therefore if <f>(x) always implies 
t/f(x ), then <f> (Socrates) implies 0 (Socrates)? and don’t you see 
that I can put 4 x is a man’ for 4 0#’ and l x is mortal’ for 4 0#’? 
And don’t you see that this proves my point?” A pupil who could 
follow this but not the original syllogism would be a psychological 
monstrosity. And even if there were such a pupil, he would still 
have to “see” the steps of my new argument. 

It follows that, when an argument is stated as simply as possible, 
the connection asserted in every step has to be a datum. But it 
is impossible that the connection in every step should have the 
highest degree of credibility, because even the best mathematicians 
sometimes mate mistakes. In fact, our perceptions of the logical 
connections between propositions, like our sense-perceptions and 
our memories, can be ordered by their degrees of credibility: 
in some, we see the logical connection so clearly that we cannot 
be made to doubt it, while in others our perception of the con- 
nection is so faint that we are not sure whether we see it or not. 

I shall henceforth assume that a datum, in the sense defined 
at the beginning of this section, may be uncertain in a greater 
or less degree. We can, theoretically, make a connection between 
this kind of uncertainty and the kind derived from mathematical 
probability, if we suppose that an uncertainty of one kind can 
be judged greater than, equal to, or less than, one of the other 
kind. For example, when I think I hear a faint sound, but am 
not sure, I may theoretically be able to say: The occurrence of 
this sound has the same degree of rational credibility as the 
occurrence of double sixes with dice. In some degree, such com- 
parisons could be tested, by collecting evidence of mistakes as 
to faint sensations and working out their frequency. All this is 
vague, and I do not see how to make it precise. But at any rate 
it suggests that the uncertainty of data is quantitative, and can 
be equal or unequal to the uncertainty derived from a probability 
inference. I shall assume this to be the case, while admitting that, 
in practice, the numerical measurement of the uncertainty of a 
datum is seldom possible. We may say that the uncertainty is a 
half when the doubt is such as to leave an even balance between 
belief and disbelief. But such a balance can only be established 
by introspection, and is incapable of being confirmed by any sort 
of test. 



The admission of uncertainty in data complicates the process 
of estimating the rational credibility of a proposition. Let us 
suppose that a certain proposition p has a degree of credibility x 
on its own account, as a datum; and let us suppose that there 
is also a conjunction h of propositions, having intrinsic credibility jy, 
from which it follows, by an argument having credibility s, that p 
has a degree of credibility w. What, then, is the total credibility 
of p ? Perhaps we might be inclined to say that it is x + yzw. 
But h also is sure to have a derived as well as an intrinsic credibility, 
and this will increase the credibility of x . In fact, the complications 
will soon become unmanageable. This causes a certain approxi- 
mation to the theory of Hegel and Dewey. 

Given a number of propositions, each having a fairly high 
degree of intrinsic credibility, and given a system of inferences 
by virtue of which these various propositions increase each other ’s 
credibility, it may be possible in the end to arrive at a body of 
interconnected propositions having, as a whole, a very high degree 
of credibility. Within this body, some are only inferred, but none 
are only premisses, for those which are premisses are also con- 
clusions. The edifice of knowledge may be compared to a bridge 
resting on many piers, each of which not only supports the road- 
way but helps the other piers to stand firm owing to intercon- 
necting girders. The piers are the analogues of the propositions 
having some intrinsic credibility, while the upper portions of the 
bridge are the analogues of what is only inferred. But although 
each pier may be strengthened by the other piers, it is the solid 
ground that supports the whole, and in like manner it is intrinsic 
credibility that supports the whole edifice of knowledge. 


Subjective certainty is a psychological concept, while credibility 
is at least in part logical. The question whether there is any 
connection between them is a form of the question whether we 
know anything. Such a question cannot be discussed on a basis 
of complete scepticism; unless we are prepared to assert something , 
no argument is possible. 

Let us first distinguish three kinds of certainty . 

(i) A propositional function is certain with respect to another 
when the class of terms satisfying the second is part of the class 


HUMAN knowledge: its scope and limits 

of terms satisfying the first. E.g. “x is an animal” is certain in 
relation to “ x is a rational animal”. This meaning of certainty 
belongs to mathematical probability. We will call this kind of 
certainty “logical”. 

(2) A proposition is certain when it has the highest degree of 
credibility, either intrinsically or as a result of argument. Perhaps 
no proposition is certain in this sense, i.e. however certain it 
may be in relation to a given person’s knowledge, further know- 
ledge might increase its degree of credibility. We will call this 
kind of certainty “epistemological”. 

(3) A person is certain of a proposition when he feels no doubt 
whatever of its truth. This is a purely psychological concept, and 
we will call it “psychological certainty”. 

Short of subjective certainty, a man may be more or less con- 
vinced of something. We feel sure that the sun will rise to-morrow, 
and that Napoleon existed; we are less sure of quantum theory 
and the existence of Zoroaster; still less sure that Eddington got 
the number of electrons exactly right, or that there was a king 
called Agamemnon at the siege of Troy. These are matters as 
to which there is fairly general agreement, but there are other 
matters as to which disagreement is the rule. Some people feel 
no doubt that Churchill is good and Stalin bad, others think the 
opposite; some people were utterly certain that God was on the 
side of the Allies, others thought that He was on the side of the 
Germans. Subjective certainty, therefore, is no guarantee of truth, 
or even of a high degree of credibility. 

Error is not only the absolute error of believing what is false, 
but also the quantitative error of believing more or less strongly 
than is warranted by the degree of credibility properly attaching 
to the proposition believed in relation to the believer’s knowledge. 
A man who is quite convinced that a certain horse will win the 
Derby is in error even if the horse does win. 

Scientific method, broadly speaking, consists of techniques and 
rules designed to make degrees of belief coincide as nearly as 
possible with degrees of credibility. We cannot, however, begin 
to seek such a harmony unless we can start from propositions 
which are both epistemologically credible and subjectively nearly 
certain. This suggests a Cartesian scrutiny, but one which, if it 
is to be fruitful, must have some non-sceptical guiding principle. 
Jf there were no relation at all between credibility and subjective 



certainty, there could be no such thing as knowledge. We assume 
in practice that a class of beliefs may be regarded as true if (a) they 
are firmly believed by all who have carefully considered them, 
(b) there is no positive argument against them, (^) there is no 
known reason for supposing that mankind would believe them 
if they were untrue. On this basis, it is generally held that judg- 
ments of perception on the one hand, and logic and mathematics 
on the other, contain what is most certain in our knowledge. We 
shall see that, if we are to arrive at science, logic and mathematics 
will have to be supplemented by certain extra-logical principles, 
of which induction has hitherto (I think mistakenly) been the 
one most generally recognized. These extra-logical principles 
raise problems which it will be our business to investigate. 

Perfect rationality consists, not in believing what is true, but 
in attaching to every proposition a degree of belief corresponding 
to its degree of credibility. In regard to empirical propositions, 
the degree of credibility changes when fresh evidence accrues. 
In mathematics, the rational man who is not a mathematician 
will believe what he is told; he will therefore change his beliefs 
when mathematicians discover errors in the work of their pre- 
decessors. The mathematician himself may be completely rational 
in spite of making a mistake, if the mistake is one which at the 
time is very difficult to detect. 

Whether we ought to aim at rationality is an ethical question. 
I shall consider some aspects of it in the following section. 


Bishop Butler’s statement that probability is the guide of life 
is very familiar. Let us consider briefly what it can mean, how 
far it is true, and what is involved in believing it to have the 
degree of truth that it seems to possess. 

Most ethical theories are of one of two kinds. According to 
the first kind, good conduct is conduct obeying certain rules; 
according to the second, it is conduct designed to realize certain 
ends. There are theories which are of neither of these two kinds, 
but for our purposes we may ignore them. 

The first type of theory is exemplified by Kant and the 
Decalogue. The Decalogue, it is true, is not a pure example of 
this type of theory, since reasons are given for some of the com- 


human knowledge: its scope and limits 

mandments. You must not worship graven images, because God 
will be jealous; you should honour your parents, because it 
diminishes your chances of death. It is of course easy to find 
reasons against murder and theft, but none are given in the Ten 
Commandments. If reasons are given, there will be exceptions, 
and common sense has in general recognized them, but none are 
admitted in the text. 

When ethics is considered to consist of rules of conduct, 
probability plays no part in it. It is only in the second type of 
ethical theory, that in which virtue consists in aiming at certain 
ends, that probability is relevant. So far as the relation to proba- 
bility is concerned, it makes very little difference what end is 
chosen. For the sake of definiteness, let us suppose the end to 
be the greatest possible excess of pleasure over pain, a pleasure 
and a pain being considered equal when a person who has the 
choice is indifferent whether he has both or neither. We may 
designate this end briefly as that of maximizing pleasure. 

We cannot say that the virtuous man will act in the way that 
will in fact maximize pleasure, since he may have no reason to 
expect this result. It would have been a good thing if Hitler’s 
mother had killed him in infancy, but she could not know this. 
We must therefore say that the virtuous man will act in the way 
which, so far as his knowledge goes, will probably maximize 
pleasure. The kind of probability that is involved is obviously 
degree of credibility. 

The probabilities concerned are to be estimated by the rules 
for computing “expectation”. That is to say, if there is a proba- 
bility p that a certain act will have among its consequences a 
pleasure of magnitude x , this contributes an amount p x to the 
expectation. Since distant consequences seldom have any appre- 
ciable probability, this justifies the practical man in usually con- 
fining his attention to the less remote consequences of his action. 

There is another consideration: the calculations involved are 
often difficult, and are most difficult when the felicific properties 
of two possible actions are nearly equal, in which case the choice 
is unimportant. Therefore as a rule it is not worth while to 
determine with any care which action will produce the most 
pleasure. This is the reason in favour of rules of action, even if 
our ultimate ethic rejects them: they can be right in the great 
majority of cases, and save us the trouble and waste of time 



involved in estimating probable effects. But the rules of action 
themselves should be carefully justified by their felicific character, 
and where really important decisions are concerned it may be 
necessary to remember that the rules are not absolute. Currency 
reform usually involves something like theft, and war involves 
killing. The statesman who has to decide whether to reform the 
currency or to declare war has to go behind rules and do his best 
to estimate probable consequences. It is only in this sense that 
probability can be the guide of life, and that only in certain 

There is, however, another and humbler sense of the dictum, 
which was perhaps that intended by the Bishop. This is, that 
we should, in practice, treat as certain whatever has a very high 
degree of probability. This is merely a matter of common sense, 
and raises no issue that is of interest to the theory of probability. 



Chapter VII 



T he problem of induction is a complex one, having various 
aspects and branches. I shall begin by stating the problem 
of induction by simple enumeration. 

I. The fundamental question, to which others are subsidiary, 
is this: Given that a number of instances of a class a have all 
been found to belong to a class 0, does this make it probable, 
(a) that the next instance of a will be a /3, or ( b ) that all a’s are j3’s ? 

II. If either of these is not true universally, are there dis- 
coverable limitations on a and f} which make it true ? 

III. If either is true with suitable limitations, is it, when so 
limited, a law of logic or a law of nature ? 

IV. Is it derivable from some other principle, such as natural 
kinds, or Keynes's limitation of variety, or the reign of law, or 
the uniformity of nature, or what not ? 

V. Should the principle of induction be stated in a different 
form, viz.: Given a hypothesis h which has many known true 
consequences and no known false ones, does this fact make h 
probable ? And if not generally, does it do so in suitable circum- 
stances ? 

VI. What is the minimum form of the inductive postulate 
which will, if true, validate accepted scientific inferences? 

VII. Is there any reason, and if so what, to suppose this 
minimum postulate true ? Or, if there is no such reason, is there 
nevertheless reason to act as if it were true ? 

There is need, in these discussions, to remember the ambiguity 
in the word “probable" as commonly used. When I say that, in 
certain circumstances, “probably" the next a will be a #, I shall 
hope to be able to interpret this according to the finite frequency 
theory. But if I say that the inductive principle is “probably" 
true, I shall have to be using the word “probably" to express 
a high degree of credibility. Confusions may easily arise through 
not keeping these two meanings of the word “probable" suffi- 
ciently separate. 



The discussions upon which we shall be engaged have a history 
which may be considered to begin with Hume. On a large number 
of subsidiary points definite results have been obtained ; sometimes 
these points were not recognized, at first, to be subsidiary. But 
investigation has made it, by now, fairly clear that the technical 
discussions which reach results throw little light on the main 
problem, which remains substantially as Hume left it. 


Induction by simple enumeration is the following principle: 
“Given a number n of a’s which have been found to be 0’s, 
and no a which has been found to be not a /}, then the two 
statements: (a) ‘the next a will be a ( b ) ‘all a’s are £V, both 
have a probability which increases as n increases, and approaches 
certainty as a limit as n approaches infinity.” 

I shall call (a) “particular induction” and (b) “general 
induction”. Thus (a) will argue from our knowledge of the past 
mortality of human beings that probably Mr. So-and-So will 
die, whereas (6) will argue that probably all men are mortal. 

Before proceeding to more difficult or doubtful points, there 
are some rather important questions which can be decided without 
great difficulty. These are: 

(1) If induction is to serve the purposes which we expect it to 
serve in science, “probability” must be so interpreted that a 
probability-statement asserts a fact ; this requires that the kind of 
probability involved should be derivative from truth and false- 
hood, not an indefinable; and this, in turn, makes the finite- 
frequency interpretation more or less inevitable. 

(2) Induction appears to be invalid as applied to the series of 
natural numbers. 

(3) Induction is not valid as a logical principle. 

(4) Induction requires that the instances upon which it is based 
should be given as a series, not merely as a class. 

(5) Whatever limitation may be necessary to make the principle 
valid must be stated in terms of the intensions by which the classes 
a and )8 are defined, not in terms of extensions. 

(6) If the number of things in the universe is finite, or if some 
finite class is alone relevant to the induction, then induction, for a 
sufficient n, becomes demonstrable; but in practice this is un- 


HUMAN knowledge: its scope and limits 

important, because the n concerned would have to be larger than 
it ever can be in any actual investigation. 

I shall now proceed to prove these propositions. 

(1) If “probability” is taken as an indefinable, we are obliged 
to admit that the improbable may happen, and that, therefore, a 
probability-proposition tells us nothing about the course of 
nature. If this view is adopted, the inductive principle may be 
valid, and yet every inference made in accordance with it may 
turn out to be false; this is improbable, but not impossible. 
Consequently a world in which induction is true is empirically 
indistinguishable from one in which it is false. It follows that 
there can never be any evidence for or against the principle, and 
that it cannot help us to infer what will happen. If the principle 
is to serve its purpose, we must interpret “probable” as meaning 
“what in fact usually happens”; that is to say, we must interpret 
a probability as a frequency. 

(2) Induction in arithmetic . — It is easy in arithmetic to give 
examples of inductions which lead to true conclusions, and others 
which lead to false conclusions. Jevons gives the two instances: 

5. 15. 35. 45. 6 5. 95 
7. 17. 37. 47. 67, 97 

In the first row, every number ends in 5 and is divisible by 5 ; 
this may lead to the conjecture that every number that ends in 
5 is divisible by 5, which is true. In the second row, every number 
ends in 7 and is prime ; this might lead to the conjecture that every 
number ending in 7 is prime, which would be false. 

Or take: “Every even integer is the sum of 2 primes”. This is 
true in every case in which it has been tested, and the number of 
such cases is enormous. Nevertheless there remains a reasonable 
doubt as to whether it is always true. 

As a striking example of failure of induction in arithmetic, take 
the following : l 

Put tt(x) = number of primes < x 

li(*) = 

, it 
log t 

1 See Hardy, Ramanujan t pp. 16, 17. 


It is known that, when x is large, tt(x) and li(jc) are nearly equal. 
It is also known that, for every known prime, 

rr(x) < li(jc) 

Gauss conjectured that this inequality always holds. It has been 
tested for all primes up to io 7 and for a good many beyond this, 
and no particular case of its falsity has been discovered. Never- 
theless Littlewood proved in 1912 that there are an infinite 
number of primes for which it is false, and Skewes ( L.M.S . 
Journal , 1933) proved that it is false for some number less than 





It will be seen that Gauss’s conjecture, though it turned out to 
be false, had in its favour vastly better inductive evidence than 
exists for even our most firmly rooted empirical generalizations. 

Without going so deeply into the theory of numbers, it is easy 
to construct false inductions in arithmetic in any required number. 
For instance, no number less than n is divisible by n. We can 
make n as large as we please, and thus obtain as much inductive 
evidence as we choose in favour of the generalization: “No 
number is divisible by n”. 

It is obvious that any n integers must possess many common 
properties which most integers do not possess. For one thing, if 
m is the greatest of them, they all possess the infinitely rare 
property of being not greater than m. There is therefore no 
validity in either a general or a particular induction as applied to 
integers, unless the property to which induction is to be applied 
is somehow limited. I do not know how to state such a limitation, 
and yet any good mathematician will have a feeling, analogous to 
common sense, as to the sort of property that is likely to allow 
an induction which turns out to be valid. If you have noticed that 
I + 3 = I + 3 + 5 = 3 2 . I + 3 + 5 + 7 = 4 2 . y° u will be 
inclined to conjecture that 

1 + 3 + 5 + • • • + ( 2n - J ) = * 2 ' 

and this conjecture can easily be proved correct. Similarly 
if you have noticed that i 8 + 2 s = 3 2 , i 8 + 2 s + 3* = 6 2 , 


human knowledge: its scope and limits 

i 8 + 2 8 + 3 8 + 4 s = io 2 , you may conjecture that the sum of 
the first n cubes is always a square number, and this again is easily 
proved. Mathematical intuition is by no means infallible as 
regards such inductions, but in the case of good mathematicians 
it seems to be oftener right than wrong. I do not know how to 
make explicit what guides mathematical intuition in such cases. 
Meanwhile, we can only say that no known limitation will make 
induction valid as applied to the natural numbers. 

(3) Induction invalid as a logical principle . — It is obvious that, 
if we are allowed to select our class f 3 as we choose, we can easily 
make sure that our induction shall fail. Let a ly a 2y . . . a n be the 
hitherto observed members of a, all of which have been found to 
be members of j8, and let « M+1 be the next member of a. So 
far as pure logic is concerned, might consist only of the terms 
a ly a 2 . . . a n ; or it might consist of everything in the universe 
except a n+1 \ or it might consist of any class intermediate 
between these two. In any of these cases the induction to a n+l 
would be false. 

It is obvious (an objector may say) that /} must not be what 
might be called a “manufactured” class, i.e. one defined partly 
by extension. In the sort of cases contemplated in inductive 
inference, )8 is always a class known in intension, but not in 
extension except as regards the observed members a ly a 2 . . a n 
and such other members of j8, not members of a, as may happen 
to have been observed. 

It is very easy to make up obviously invalid inductions. A rustic 
might say: all the cattle I have ever seen were in Herefordshire, 
therefore probably all cattle are in that county. Or we might argue : 
No man now alive has died, therefore probably all the men now 
alive are immortal. The fallacies in such inductions are fairly 
obvious, but they would not be fallacies if induction were a 
purely logical principle. 

It is therefore clear that, if induction is to be not demonstrably 
false, the class p must have certain characteristics, or be related 
in some specific way to the class a. I am not contending that with 
these limitations the principle must be true ; I am contending that 
without them it must be false. 

(4) In empirical material instances come in a time-order, and 
therefore are always serial. When we consider whether induction 
is applicable in arithmetic, we naturally think of the numbers as 



arranged in order of magnitude. But if we are allowed to arrange 
them as we like, we can obtain strange results ; for instance, as we 
saw, we can prove that it is infinitely improbable that a number 
chosen at haphazard will not be a prime. 

It is essential to the enunciation of particular induction that 
there should be a next instance, which demands a serial arrange- 

If there is to be any plausibility about general induction, we 
must be given that the first n members of a are found to be 
members of /?, not merely that a and B have n members in 
common. This again requires a serial arrangement. 

( 5 ) Assuming it admitted that, if an inductive inference is to 
be valid, there must be some relation between a and /3, or some 
characteristic of one of them, in virtue of which it is valid, it is 
clear that this relation must be between intensions , e.g. between 
“human” and “mortal”, or between “ruminant” and “dividing 
the hoof”. We seek to infer an extensional relation, but we do 
not originally know the extensions of a and j 8 when we are dealing 
with empirically given classes of which new members become 
known from time to time. Everyone would admit “dogs bark” as 
a good induction; we expect a correlation between the visual 
appearance of an animal and the noise it makes. This expectation 
is, of course, the result of another, wider induction, but that is 
not at the moment the point that concerns me. What concerns me 
is the correlation of a kind of shape with a kind of noise, both 
intensions, and the fact that certain intensions seem to us more 
likely to be inductively related than certain others. 

( 6 ) This point is obvious. If the universe is finite, complete 
enumeration is theoretically possible, and before it has been 
achieved the ordinary calculus of probability shows that an 
induction is probably valid. But in practice this consideration has 
no importance, because of the disproportion between the number 
of things we can observe and the number of things in the universe. 

Let us now revert to the general principle, remembering that 
we have to seek some limitation which will make it possibly valid. 
Take particular induction first. This says that, if a random 
selection of n members of a is found to consist wholly of members 
of £, it is probable that the next member of a will be & /?, i.e. most 
of the remaining as are £’s. This itself need only be probable. 
We may suppose a to be a finite class, containing (say) N members. 



Of these we know that at least n are members of /J. If the total 
number of members of a that are members of /S is m, the total 

number of ways of selecting n terms is and the 

total number of ways of selecting n terms that are a’s is 


n ! (m — n ) ! 
of a’s is 

Therefore the chance of a selection consisting wholly 

m ! (N — n ) ! 

N ! (m — n ) ! 

If A. is the a priori likelihood of m being the number of terms 
common to a and j8, then the likelihood after experience is 

ml (N — ri)\ /v~» ml (N — n)\ 

K ' N! (m-n)l / Aj Pm ' N! (tn - nj\ 

Let us call this q m . 

If the number of members common to a and j3 is m, then after 
withdrawing n a’s that are j8’s there remain m — n jS’s and N — n 
not - jS’s. Therefore, from the hypothesis that a and # have m 

m — n 

members in common, we get a probability q m . of another j8. 

Therefore the total probability is 

S m — n 

q m N 

The value of this depends entirely on the p m ’ s, which there is no 
valid way of estimating. If we assume with Laplace that every 
value of m is equally probable we get Laplace’s result, that the 

n + i 

chance of the next a being a B is — : — . If we assume that, a 

priori , each a is equally likely to be a jS and not be a /?, we get 
the value i /2. Even with Laplace’s hypothesis th c general induction 
n + i 

has only a probability -, which is usually small. 

We need therefore some hypothesis which makes p m large when 
m is nearly N . This will have to depend upon the nature of the 
classes a and jS if it is to have any chance of validity. 

1 I” means the product of all whole numbers from i to N. 




From the time of Laplace onward, various attempts have been 
made to show that the probable truth of an inductive inference 
follows from the mathematical theory of probability. It is now 
generally agreed that these attempts were all unsuccessful, and 
that, if inductive arguments are to be valid, it must be in virtue 
of some extra-logical characteristic of the actual world, as opposed 
to the various logically possible worlds that may be contemplated 
by the logician. 

The first of such arguments is due to Laplace. In its valid, 
purely mathematical, form, it is as follows: 

There are n + i bags, similar in external appearance, and each 
containing n balls. In the first, all the balls are black; in the 
second, one is white and the rest black; in the (r + i) th , r are 
white and the rest black. One of these, of which the composition 
is unknown, is selected, and m balls are withdrawn from it. They 
prove to be all white. What is the probability {a) that the next 
ball drawn will be white, (b) that we have chosen the bag con- 
sisting wholly of white balls? 

The answer is : (a) the chance that the next ball will be white is 
m -j- i 

-; ( h ) the chance that we have chosen the bag in which all 

m + 2’ 
the balls are white is 

m + i 
n + i * 

This valid result has a straightforward interpretation on the 
finite frequency theory. But Laplace infers that if m A’s have 
been found to be B’s, the chance that the next A will be a B is 

m + i m + i 

— ; — , and the chance that all A’s are B’s is — ; — . He gets this 
m -f- 2 n -j- i 

result by assuming that, given n objects of which we know nothing, 
the probabilities that o, i, 2 , ... n of them are B’s are all equal. 
This, of course, is an absurd assumption. If we replace it by the 
slightly less absurd assumption that each of the objects has an 
equal chance of being a B or not a B, the chance that the next A 
will be a B remains 1/2, however many A’s have been found to 
be B’s. 

Even if his argument were accepted, the general induction 
remains improbable if n is much greater than m, though the 


HUMAN knowledge: its scope and limits 

particular induction may become highly probable. In fact, how- 
ever, his argument is only a historical curiosity. ^ 

Keynes, in his Treatise on Probability , has done the best that 
can be done for induction on purely mathematical lines, and has 
decided that it is inadequate. His result is as follows. 

Let j be a generalization, x ly x 2 , . . . observed instances 
favourable to it, and h the general circumstances so far as they 
are relevant. 


x ] /h — x 2 lh = etc. 
Pn = gl h X l X 2 • • • X 

Thus p n is the probability of the general induction after n favour- 
able instances. Write g for the negation of g y and p 0 for gjh , the 
a priori probability of the generalization. 

Ptl Po+ *1 *2 ■ • X J£ H* ~ Po) 

As n increases, this approaches i as a limit if 

*1 *2 • • *Jg k 

approaches zero as a limit; and this happens if there are finite 
quantities e and rj such that, for all sufficiently great r* s, 

x r /x x x 2 . . g h < i — € and p 0 > rj 

Let us consider these two conditions. The first says that there 
is a quantity i — e, less than i, such that, if the generalization is 
false, the probability of the next instance being favourable will 
always, after a certain number of favourable instances, be less 
than this. Consider, as an instance of its failure, the generalization 
“all numbers are non-prime , \ As we move up the number-series, 
primes become rarer, and the chance of the next number after r 
non-primes being itself a non-prime increases, and approaches 
certainty as a limit if r is kept constant. This condition, therefore, 
may fail. 

But the second condition, that g must, antecedently to the 
beginning of the induction, have a probability greater than some 
finite probability, is more difficult. In general, it is hard to see 
any way in which this probability can be estimated. What would 



be the probability of “all swans are white” for a person who had 
never seen a swan or been told anything about the colour of 
swans? Such questions are both obscure and vague, and Keynes 
recognizes that they make his result unsatisfactory. 1 

There is one simple hypothesis which would give the finite 
probability that Keynes wants. Let us suppose that the number 
of things in the universe is finite, say N. Let /} be a class of n 
things, and let a be a random selection of m things. Then the 
number of possible a’s is 


m !(N — m) f 

and the number of these that are contained in 8 is 


m \(n — m ) ! 

Therefore the chance of “all a’s are 0’s” is 

w!(N — 7 ri)\ 

N !(n — m ) ! 

which is finite. That is to say, every generalization as to which 
we have no evidence has a finite chance of being true. 

I fear, however, that, if N is as large as Eddington maintained, 
the number of favourable instances required to make an inductive 
generalization probable in any high degree would be far in excess 
of what is practically attainable. This way of escape, therefore, 
while excellent in theory, will not serve to justify scientific practice. 

Induction in the advanced sciences proceeds on a somewhat 
different system from that of simple enumeration. There is first 
a body of observed facts, then a general theory consistent with 
them all, and then inferences from the theory which subsequent 
observation confirms or confutes. The argument here depends 
upon the principle of inverse probability. Let p be a general 
theory, h the previously known data, and q a new experimental 
datum relevant to p . Then 

, (Plh).(q/ph) 


1 I shall return to this subject in Part VI, Chapter II. 


In the most important case, q follows from p and A, so that 
q Ip A = i. In this case, therefore, 

p/qh = 



It follows that, if qjh is very small, the verification of q greatly 
increases the probability of p. This, however, does not have quite 
the consequences one might hope. We have, putting “p” for 

qjh = pqjh + pqjh = p/h + pqjh 

because, given A, p implies q. Thus if 

y = 



we have plq A = — ; — 

i +y 

This will be a high probability if y is small. Now two circum- 
stances may make y small: (i) if p/h is large, (2) if p qjh is small, 
i.e. if q would be improbable if p were false. The difficulties in 
the way of estimating these two factors are much the same as 
those that appear in Keynes’s discussion. To obtain an estimate 
of />/A, we shall need some way of evaluating the probability of 
p antecedently to the special evidence that has suggested it, and 
it is not easy to see how this can be done. The only thing that 
seems clear is that, if a suggested law is to have an appreciable 
probability antecedently to any evidence in its favour, that must 
be in virtue of a principle to the effect that some fairly simple law 
is bound to be true. But this is a difficult matter, to which I shall 
return at a later stage. 

The probability of pq/h, in certain kinds of cases, is more 
possible to estimate approximately. Let us take the case of the 
discovery of Neptune. In this case, p is the law of gravitation, 
A is the observations of planetary motions before the discovery of 
Neptune, and q is the existence of Neptune at the place where 
calculations showed that it should be. Thus p qjh is the probability 
that Neptune would be where it was, given that the law of gravita- 
tion was false. Here we must make a proviso as to the sense in 
which we should use the word “false”. It would not be right to 
take Einstein’s theory as showing Newton’s to be “false” in the 



relevant sense. All quantitative scientific theories, when asserted, 
should be asserted with a margin of error; when this is done, 
Newton’s theory of gravitation remains true of planetary motions. 

The following argument looks hopeful, but is not in fact valid. 

In our case, apart from p or some other general law, h is 
irrelevant to q y that is to say, observations of other planets make 
the existence of Neptune neither more nor less probable than it 
was before. As for other laws, Bode’s law might be held to make 
it more or less probable that there would be a planet having more 
or less the orbit of Neptune, but would not indicate the part of 
its orbit that it had reached at a given date. If we suppose that 
Bode’s law, and any other relevant law except gravitation, con- 
ferred a probability x on the hypothesis of a planet roughly in the 
orbit of Neptune, and suppose the apparent position of Neptune 
was calculated with a margin of error 0, then the probability of 
Neptune being found where it was would be 0/27 r. Now 0 was 
very small, and it cannot be maintained that x was large. There- 
fore p q/h, which was x 0/ 27r, was certainly very small. Suppose 
we take x to be i/io and 0 to be 6 minutes, then 

_ i i_ 

^ — io 3,600 36,000 

Therefore if we suppose that pjh = 1 /36, we shall havey = 1/1000 


plqh = 

1 .001 

Thus even if, before the discovery of Neptune, the law of gravita- 
tion was as improbable as double sixes with dice, it had afterwards 
odds of 1000 to 1 in its favour. 

This argument, extended to all the observed facts of planetary 
motions, apparently shows that, if the law of gravitation had even 
a very small probability at the time when it was first enunciated, 
it soon became virtually certain. But it does nothing to help us 
to gauge this initial probability, and therefore would fail, even if 
valid, to give us a firm basis for the theoretical inference from 
observation to theory. 

Moreover the above argument is open to objection, in view of 
the fact that the law of gravitation is not the only law which would 
lead to the expectation of Neptune being where it was. Suppose 
the law of gravitation to have been true until the time /, where t 
is any moment subsequent to the discovery of Neptune ; then we 


HUMAN knowledge: its scope and limits 

should still have q/p'h = i , where/)' is the hypothesis that the law 
was true only until the time t. There was therefore better reason 
to expect the finding of Neptune than would result from pure 
chance, or from this together with Bode’s law. What was rendered 
highly probable was that the law had held until then. To infer 
that it would hold in future required a principle not derivable 
from anything in the mathematical theory of probability. This 
consideration destroys the whole force of the inductive argument 
for general theories, unless the argument is reinforced by some 
principle such as the uniformity of nature is supposed to be. 
Here again, we find that induction needs the support of some 
extra-logical general principle not based upon experience. 

d. reichenbach’s theory 

The peculiarity of Reichenbach’s theory of probability is that 
induction is involved in the very definition of a probability. His 
theory is as follows (somewhat simplified). 

Given a statistical series, e.g. such as in vital statistics, and 
given two overlapping classes a and j 8 to which some members 
of the series belong, we often find that, when the number of items 
is large, the percentage of members of j3 that are members of a 
remains approximately constant. Suppose that, when the number 
of items exceeds (say) 10,000, it is found that the proportion of 
recorded £’s that are as is never far from m/n , and that this 
rational fraction is nearer the average observed proportion than 
any other. We then “posit” that, however far the series may be 
prolonged, the proportion will always remain nearly m/n . We 
define the probability of a /8 being an a as the limit of the observed 
frequency when the number of observations is indefinitely 
increased, and in virtue of our “posit” we assume that this limit 
exists and is in the neighbourhood of m/n , where m/n is the 
observed frequency in the largest obtainable sample. 

Reichenbach asserts with emphasis that no proposition is 
certain; all are only probable in varying degrees, and every 
probability is the limit of a frequency. He admits that, in con- 
sequence of this doctrine, the items by means of which the 
frequency is computed are themselves only probable. Take e.g. 
the death rate: when a man is judged dead, he may be still alive, 
therefore every item in mortality statistics is doubtful. This 



means, by definition, that the record of a death must be one of a 
series of records, some correct, some erroneous. But those we take 
to be correct are only probably correc